Exponential functions
•Download as PPT, PDF•
0 likes•461 views
The document defines exponential functions as functions of the form f(x) = ax, where a is the base and a > 0, a ≠ 1. Exponential functions with bases greater than 1 have graphs that increase rapidly and have a horizontal asymptote of y = 0. Exponential functions with bases between 0 and 1 have graphs that increase more slowly and also have a horizontal asymptote of y = 0. Examples are given of sketching the graphs of various exponential functions, including translations and reflections of f(x) = 2x. The irrational number e, which is approximately 2.718, is important in applications involving growth and decay.
Report
Share
Report
Share
Recommended
Dividing Polynomials Slide Share
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Factor theorem
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
3 1 Quadratic Functions
This document discusses quadratic functions and graphs, including:
- Graphing quadratic functions in standard and vertex forms
- Identifying minimums, maximums, and axes of symmetry from graphs
- Finding minimums and maximums without graphing using calculators
- Applications of quadratic functions like quadratic regression and maximizing enclosed areas with fencing
The remainder theorem powerpoint
The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.
Lecture complex fractions
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
Factoring Quadratic Trinomials
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
A25-7 Quadratic Inequalities
This document provides instructions for graphing and solving quadratic inequalities. It explains how to graph parabolas defined by inequalities using a graphing calculator. It also describes how to graph systems of quadratic inequalities by first graphing the individual parabolas and then shading the overlapping regions indicated by the inequalities. When solving inequalities, it advises finding the critical values by setting the expression equal to zero and factoring, and then plotting these values on a number line to determine the solution intervals.
Inverse functions
The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
Recommended
Dividing Polynomials Slide Share
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Factor theorem
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
3 1 Quadratic Functions
This document discusses quadratic functions and graphs, including:
- Graphing quadratic functions in standard and vertex forms
- Identifying minimums, maximums, and axes of symmetry from graphs
- Finding minimums and maximums without graphing using calculators
- Applications of quadratic functions like quadratic regression and maximizing enclosed areas with fencing
The remainder theorem powerpoint
The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.
Lecture complex fractions
This document discusses simplifying complex rational expressions, which have numerators or denominators containing fractions. It provides two methods for simplification:
1) Multiplying the numerator and denominator by the lowest common denominator to clear fractions. Examples and steps are shown.
2) Dividing the numerator by the denominator after collecting like terms. An example problem is worked through to demonstrate the process. Objectives and learning outcomes are stated to guide readers.
Factoring Quadratic Trinomials
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
A25-7 Quadratic Inequalities
This document provides instructions for graphing and solving quadratic inequalities. It explains how to graph parabolas defined by inequalities using a graphing calculator. It also describes how to graph systems of quadratic inequalities by first graphing the individual parabolas and then shading the overlapping regions indicated by the inequalities. When solving inequalities, it advises finding the critical values by setting the expression equal to zero and factoring, and then plotting these values on a number line to determine the solution intervals.
Inverse functions
The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
Polynomial functions
This document discusses polynomial functions. A polynomial is a sum of monomials, and a polynomial function with a single variable is called a polynomial function. The degree of a polynomial is the highest exponent of its variable, and the leading coefficient is the coefficient of the term with the highest degree. Polynomial functions are classified based on their degree as linear, quadratic, cubic, etc. The document also discusses evaluating polynomial functions for different inputs, the general shapes of polynomial functions based on degree and leading coefficient, and the remainder and factor theorems.
Factoring the difference of two squares
The document discusses factoring the difference of two squares through examples such as (x+5)(x-5)=x^2 - 25. It explains that to factor a difference of two squares, we write the expression as the difference of two terms squared, then group the terms with the same bases and opposite signs inside parentheses. Several practice problems are provided to reinforce this technique for factoring completely the difference of two squares.
Polynomial Function and Synthetic Division
This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.
Completing the square
This document provides examples and instructions for solving quadratic equations by completing the square. It begins with examples of solving quadratic equations using the square root property. It then explains how to complete the square to write a quadratic expression as a perfect square trinomial. Examples are provided to demonstrate completing the square and using it to solve quadratic equations. The document ensures readers understand completing the square through check examples and objectives.
3.3 Zeros of Polynomial Functions
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
Multiplying polynomials
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
Polynomial equations
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
Quadratic inequalities
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
First Quarter - Chapter 2 - Quadratic Equation
Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
* The initial velocity (v) is now 20 feet/second
* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
0 = -16t^2 + 20t + 18 (subtract 20 from both sides)
Evaluating the discriminant:
(20)^2 - 4(-16)(-18) = 400 - 288 = 112
Since the discriminant is positive
QUADRATIC FUNCTIONS
The document provides an overview of quadratic functions including definitions of key terms like quadratic function, parabola, quadratic equation, and vertex form. It discusses how to graph quadratic functions, solve quadratic equations using methods like the quadratic formula, and find the maximums and minimums of quadratic functions. It includes examples of using these techniques to solve word problems involving quadratic applications. The document aims to teach students about the basic concepts of quadratic functions.
Distance formula
This document explains the distance formula and how to use it to calculate the distance between points on a Cartesian plane. It provides the steps to use the Pythagorean theorem to derive the distance formula: take the difference between the x-coordinates squared and add it to the difference between the y-coordinates squared, and take the square root of the result. It then works through an example of using the distance formula to calculate the distance between points (3,2) and (8,7), which equals 5 units. The document concludes with practice problems and assignments applying the distance formula.
5 4 function notation
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
16.1 Solving Quadratics by square roots
This document introduces the square root property and method for solving quadratic equations using square roots. It provides 3 steps: 1) isolate the term being squared, 2) take the square root of both sides, and 3) solve the remaining equation if needed. Examples are shown of solving equations of the form x^2=a, (x-b)^2=c, and (ax-b)^2=c by taking the square root and solving the resulting linear equations. It is noted that some quadratic equations using this method may have no real solutions.
Ch8-2 Solving Exponential Functions
This document discusses solving exponential equations by equating exponents. It provides examples of solving equations by making the bases the same using properties of exponents, including using negative exponents to write fractional bases as powers of an integer base. The document demonstrates these steps and techniques on sample problems, then prompts the reader to solve additional exponential equations on their own and check their work.
Remainder theorem
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
8.4 logarithmic functions
Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
Factoring the Difference of Two Squares
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
Equation of a circle
Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
Modeling with Quadratics
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
Lecture 03 special products and factoring
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
2.2 exponential function and compound interest
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
Hprec5 4
This document defines common (base 10) and natural (base e) logarithms and explains how to evaluate, solve equations involving, and graph logarithmic functions. It also discusses important properties of logarithms including:
- Logarithms are the inverse functions of exponentials.
- The natural logarithm ln(x) is the logarithm with base e, where e is an important mathematical constant approximately equal to 2.718.
- The domain of ln(x) is x > 0, since it is undefined for non-positive values. The range extends from -∞ to ∞.
More Related Content
What's hot
Polynomial functions
This document discusses polynomial functions. A polynomial is a sum of monomials, and a polynomial function with a single variable is called a polynomial function. The degree of a polynomial is the highest exponent of its variable, and the leading coefficient is the coefficient of the term with the highest degree. Polynomial functions are classified based on their degree as linear, quadratic, cubic, etc. The document also discusses evaluating polynomial functions for different inputs, the general shapes of polynomial functions based on degree and leading coefficient, and the remainder and factor theorems.
Factoring the difference of two squares
The document discusses factoring the difference of two squares through examples such as (x+5)(x-5)=x^2 - 25. It explains that to factor a difference of two squares, we write the expression as the difference of two terms squared, then group the terms with the same bases and opposite signs inside parentheses. Several practice problems are provided to reinforce this technique for factoring completely the difference of two squares.
Polynomial Function and Synthetic Division
This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.
Completing the square
This document provides examples and instructions for solving quadratic equations by completing the square. It begins with examples of solving quadratic equations using the square root property. It then explains how to complete the square to write a quadratic expression as a perfect square trinomial. Examples are provided to demonstrate completing the square and using it to solve quadratic equations. The document ensures readers understand completing the square through check examples and objectives.
3.3 Zeros of Polynomial Functions
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
Multiplying polynomials
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
Polynomial equations
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
Quadratic inequalities
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
First Quarter - Chapter 2 - Quadratic Equation
Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
* The initial velocity (v) is now 20 feet/second
* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
0 = -16t^2 + 20t + 18 (subtract 20 from both sides)
Evaluating the discriminant:
(20)^2 - 4(-16)(-18) = 400 - 288 = 112
Since the discriminant is positive
QUADRATIC FUNCTIONS
The document provides an overview of quadratic functions including definitions of key terms like quadratic function, parabola, quadratic equation, and vertex form. It discusses how to graph quadratic functions, solve quadratic equations using methods like the quadratic formula, and find the maximums and minimums of quadratic functions. It includes examples of using these techniques to solve word problems involving quadratic applications. The document aims to teach students about the basic concepts of quadratic functions.
Distance formula
This document explains the distance formula and how to use it to calculate the distance between points on a Cartesian plane. It provides the steps to use the Pythagorean theorem to derive the distance formula: take the difference between the x-coordinates squared and add it to the difference between the y-coordinates squared, and take the square root of the result. It then works through an example of using the distance formula to calculate the distance between points (3,2) and (8,7), which equals 5 units. The document concludes with practice problems and assignments applying the distance formula.
5 4 function notation
This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
16.1 Solving Quadratics by square roots
This document introduces the square root property and method for solving quadratic equations using square roots. It provides 3 steps: 1) isolate the term being squared, 2) take the square root of both sides, and 3) solve the remaining equation if needed. Examples are shown of solving equations of the form x^2=a, (x-b)^2=c, and (ax-b)^2=c by taking the square root and solving the resulting linear equations. It is noted that some quadratic equations using this method may have no real solutions.
Ch8-2 Solving Exponential Functions
This document discusses solving exponential equations by equating exponents. It provides examples of solving equations by making the bases the same using properties of exponents, including using negative exponents to write fractional bases as powers of an integer base. The document demonstrates these steps and techniques on sample problems, then prompts the reader to solve additional exponential equations on their own and check their work.
Remainder theorem
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
8.4 logarithmic functions
Logarithms were invented to solve exponential equations like 2x = 6, where x is between 2 and 3. A logarithm with base b is defined as logby = x if and only if bx = y. Common properties of logarithms include logb1 = 0 and logbb = 1. Logarithmic functions have inverse relationships with exponential functions and can be graphed by shifting and reflecting the standard logarithmic curve.
Factoring the Difference of Two Squares
The document discusses factoring the difference of two squares. It involves reviewing factoring the difference of two squares, which involves recognizing that the difference of two squares can be written as the product of two binomials, where one binomial contains the sum of the two terms and the other contains their difference.
Equation of a circle
Here are some things you did well and could improve on:
WWW:
- You explained the key concepts around writing the equation of a circle clearly and concisely. Breaking it down step-by-step makes it easy to understand.
- Providing examples with worked solutions is very helpful for reinforcement. The visual diagrams additionally aid comprehension.
- Giving practice problems for students to try on their own, along with answers, allows for application of the material.
EBI:
- Some of the text could be formatted for easier reading (e.g. consistent formatting of equations).
- Adding brief summaries or recaps after sections of explanation may aid retention.
- Providing guidance on common errors
Modeling with Quadratics
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
Lecture 03 special products and factoring
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
What's hot (20)
Viewers also liked
2.2 exponential function and compound interest
The document discusses exponential functions and their properties. It defines exponential functions as functions of the form f(x) = bx where b > 0 and b ≠ 1. Some key points made in the document include:
- The rules for exponents such as b0, b-k, (√b)k, and (b1/k) are explained.
- Exponential functions are defined for all real numbers x.
- Examples are provided to illustrate calculating exponential expressions and functions with integer, fractional, decimal, and real-number exponents.
- Exponential functions appear in various fields like finance, science, and engineering. Common exponential functions mentioned are y = 10x, y = ex, and y
Hprec5 4
This document defines common (base 10) and natural (base e) logarithms and explains how to evaluate, solve equations involving, and graph logarithmic functions. It also discusses important properties of logarithms including:
- Logarithms are the inverse functions of exponentials.
- The natural logarithm ln(x) is the logarithm with base e, where e is an important mathematical constant approximately equal to 2.718.
- The domain of ln(x) is x > 0, since it is undefined for non-positive values. The range extends from -∞ to ∞.
Tutorials--Exponential Functions in Tabular and Graph Form
This document outlines 54 tutorials that provide examples of constructing tables and graphs for exponential functions of various bases (2, 10, e) and characteristics of the coefficients a and b. Each tutorial works through an example of an exponential function of the form y = a*b^x, varying the values of a and b to illustrate different patterns in the table and graph.
Lesson 3.1
This document contrasts linear and exponential functions. Linear functions increase at a constant rate, while exponential functions increase at a changing rate but at a constant percent rate. An example compares two job offers, one with linear growth and one exponential. Exponential growth is also modeled using compound interest in a savings account. The growth factor is introduced to write the compound interest formula in a simpler way. Exponential decay is demonstrated using a medication example where the amount in the bloodstream decreases by 15% each hour. Finally, general properties and graphs of exponential functions are reviewed.
Simple interest
This document provides information about simple interest, including its meaning, formula, and examples. Simple interest is interest computed only on the principal amount and not on accumulated interest from prior periods. The formula for simple interest is: SI = Principal x Rate x Time / 100. Three examples demonstrate calculating simple interest using different principal amounts, interest rates, and time periods. The document concludes with two homework questions applying the simple interest formula.
4.1 exponential functions 2
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
Indices & logarithm
Logarithms were originally developed to simplify complex arithmetic calculations by transforming multiplication into addition. A logarithm relates an exponential equation to its logarithmic form and vice versa. The key property of logarithms is that they allow rewriting exponential equations as logarithmic expressions and vice versa. This property is important for solving exponential equations. Logarithmic expressions are undefined if the argument is negative.
61 exponential functions
The document reviews the basics of exponential notation. It defines AN as A multiplied by itself N times, with A as the base and N as the exponent. The rules of exponents are then presented, including the multiplication, division, power, negative, fractional, and decimal exponent rules. Examples are provided to demonstrate simplifying expressions using these rules.
Hprec5 5
This document discusses the properties and laws of logarithms. It explains that common and natural logarithms have the same properties and laws, even though their bases differ. The key properties covered include: logarithms are only defined for positive values; ln1=0, ln e=1; and the product, quotient, and power laws. Several examples are worked through to demonstrate how to use these properties and laws to simplify logarithmic expressions and solve equations. The document emphasizes that understanding logarithmic properties and how to manipulate logarithms is important for advanced math, science, and engineering courses.
Lesson 14: Exponential Functions
This document provides an overview of exponential functions including:
- Definitions of exponential functions for positive integer, rational, and irrational exponents. Conventions are established to define exponents for all real number bases and exponents.
- Properties of exponential functions including that they are continuous functions with domain of all real numbers and range from 0 to infinity. Properties involving addition, subtraction, multiplication, and division of exponents are proven.
- Examples are provided to demonstrate simplifying expressions using properties of exponents, including fractional exponents.
Simple Interest
Krish wants to buy a house for Rs. 30,00,000 but does not have enough savings, so he must take out a loan. A loan is money borrowed from a lender like a bank that must be paid back with additional interest. Simple interest is calculated using the principal amount, interest rate, and time period, with the formula: Simple Interest = Principal * Rate * Time / 100. The document provides examples of calculating simple interest on various loans to explain the concept.
Compound Interest
Compound interest is interest calculated on both the principal amount and on any interest that has already been earned over multiple periods. It is usually used for loans of more than one year. Compound interest is higher than simple interest because the interest earns interest. The compound interest formula can be used to calculate the future value (FV) of an investment given the present value (PV), interest rate per period (i), and number of compounding periods (n). An example calculates that $1500 compounded quarterly at 6.75% annually for 10 years results in a future value of $2929.
Exponential functions
The document discusses exponential functions and exponential equations. Exponential functions have the form f(x) = bx, where b is the base and x is the exponent. These functions are important in modeling real-world phenomena like population growth. Exponential equations set the exponents of the same base equal to solve for the variable. They can be solved by rewriting all terms to have the same base, setting the exponents equal, and solving the resulting equation.
Logarithmic Functions
The document defines logarithmic functions and provides examples of converting between logarithmic and exponential forms. It discusses the properties of logarithmic equality and solving logarithmic equations by rewriting them in exponential form. Key points include: logarithmic and exponential forms are inverses; the exponent becomes the logarithm; logarithmic functions have a domain of positive real numbers; and solving logarithms involves setting arguments equal when bases are the same.
Exponential functions
An exponential function has the form y = a · bx, where a and b are constants and b must be greater than 0. This document discusses exponential functions through examples and explanations. It explores how changing the constants a and b impact the graph of the function. It also introduces the equality property of exponential functions, which states that if the bases are the same, the exponents can be set equal to solve equations. Several examples demonstrate how to use this property to solve equations involving exponential functions.
Properties of logarithms
This document discusses properties of logarithms, including:
1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents.
2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm.
3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.
Compound Interest
The document explains compound interest, how it is calculated, and provides examples. Compound interest is interest paid on both the principal amount and accumulated interest over time. It is commonly used in savings accounts and works by compounding interest periodically, such as daily, quarterly, or annually. The document shows how to use a compound interest table to calculate the balance and interest earned for various principal amounts over different time periods at given interest rates that are compounded periodically.
Simple interest
The power point explains the concept of simple interest and the development of formula of simple interest.It also attempts to solve few problems.
Exponential Functions
This document discusses exponential functions. It defines exponential functions as functions where f(x) = ax + B, where a is a real constant and B is any expression. Examples of exponential functions given are f(x) = e-x - 1 and f(x) = 2x. The document also discusses evaluating exponential equations with like and different bases using logarithms. It provides examples of graphing exponential functions and discusses the key characteristics of functions of the form f(x) = bx, including their domains, ranges, and asymptotic behavior. The document concludes with an example of applying exponential functions to model compound interest.
Exponential and logarithmic functions
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
Viewers also liked (20)
Tutorials--Exponential Functions in Tabular and Graph Form
Tutorials--Exponential Functions in Tabular and Graph Form
Similar to Exponential functions
Exponential functions (1)
This document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are discussed, including that they have a horizontal asymptote of y = 0 and increase or decrease exponentially as x increases or decreases. Examples are given of sketching the graphs of specific exponential functions, including stating their domains and ranges. The special exponential function f(x) = ex is also introduced, where e is the irrational number approximately equal to 2.718.
exponential functions and their graphs.ppt
This document discusses exponential functions and their graphs. It defines the exponential function f(x) = ax and gives examples. It shows how to evaluate exponential functions at different values of x. It explains the graphs of exponential functions with bases greater than and less than 1, and how they have horizontal asymptotes at y = 0. It provides examples of sketching graphs of exponential functions and stating their domains and ranges. It introduces the irrational number e and the natural exponential function f(x) = ex. It concludes with formulas for compound interest and an example problem.
Exponential functions
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.
3_1 Exponential functions and their graphs (1).ppt
Exponential functions, definituon and examples. Graph and characteristics of exponential functions. Applications of exponential functions.
Function
FUNCTION
A function is a set of ordered pairs which relates two sets such that to each element of one set there is only one element of the second set
Bahan ajar kalkulus integral
1. A differential equation is an equation that relates an unknown function with some of its derivatives. The document provides a step-by-step example of solving a differential equation to find the xy-equation of a curve with a given gradient condition.
2. The key steps are: (1) write the derivative term as a fraction, (2) integrate both sides, (3) apply the initial condition to determine the constant term, (4) write the final function relationship.
3. Common types of differential equations discussed are separable first order equations, where the derivative terms can be isolated by dividing both sides.
Maths 12
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
Interpolation functions
The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.
2nd-year-Math-full-Book-PB.pdf
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
2018-G12-Math-E.pdf
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
Grph quad fncts
The document discusses graphing quadratic functions. It defines a quadratic function as f(x) = ax^2 + bx + c where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic function is a parabola that is symmetrical about an axis. When the leading coefficient a is positive, the parabola opens upward and the vertex is a minimum. When a is negative, the parabola opens downward and the vertex is a maximum. Standard forms for quadratic functions and methods for finding characteristics like the vertex, axis of symmetry, and x-intercepts from the equation are also presented.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Quadratic
The document discusses quadratic functions. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c where a ≠ 0. It explains that the graph of a quadratic function is a parabola. If a>0, the parabola opens up and there is a minimum point. If a<0, the parabola opens down and there is a maximum point. The roots of the quadratic function correspond to the x-intercepts of the graph. If b^2 - 4ac > 0 there are two distinct roots, if b^2 - 4ac = 0 there is one repeated root, and if b^2 - 4ac < 0 there are no real roots
Function evaluation, termination, vertical line test etc
The document discusses functions and their inverses. It defines key terms like domain, range, and inverse functions. It provides examples of determining if a relationship is a function using the vertical line test or one-to-one mapping. The document also covers finding inverses of straight lines, parabolas, exponential graphs and logarithmic graphs. It shows how to sketch these graphs and determine their domains and ranges. Finally, it discusses solving cubic expressions through finding factors and remainders.
functions limits and continuity
This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.
Chapter 3
1) A quadratic function is an equation of the form f(x) = ax^2 + bx + c, where a ≠ 0. Its graph is a parabola.
2) The vertex of a parabola is the point where it intersects its axis of symmetry. If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
3) The standard form of a quadratic equation is f(x) = a(x - h)^2 + k, where the vertex is (h, k) and the axis of symmetry is x = h.
Math1000 section2.6
The document discusses transformations of functions including translations, reflections, stretches, and compressions. It defines even and odd functions and provides examples of applying transformations to common parent functions like quadratic, square root, and absolute value functions. Examples are worked through to demonstrate how to obtain the graph of one function from another by applying different transformations.
Math - Operations on Functions, Kinds of Functions
Math 17 lesson
If downloaded, please credit A.B.Balbuena of the Institute of Mathematics, University of the Philippines-Diliman
Quadraticfunctionpresentation 100127142417-phpapp02
The document defines quadratic functions and discusses their various forms, including general, vertex, and factored forms. It also covers solving quadratic equations using methods like the quadratic formula, factoring, and completing the square. Additionally, it discusses key features of quadratic graphs like x-intercepts, y-intercepts, the vertex, and concavity. Examples are provided to illustrate finding these features and graphing parabolas.
Modul 3 quadratic function
This document provides information about quadratic functions including:
1) The general form of a quadratic function is f(x) = ax2 + bx + c, where a, b, c are constants and a ≠ 0.
2) Characteristics of quadratic functions include involving one variable only and the highest power of the variable being 2.
3) Examples are provided to demonstrate recognizing quadratic and non-quadratic functions based on their form.
Similar to Exponential functions (20)
3_1 Exponential functions and their graphs (1).ppt
3_1 Exponential functions and their graphs (1).ppt
Function evaluation, termination, vertical line test etc
Function evaluation, termination, vertical line test etc
Math - Operations on Functions, Kinds of Functions
Math - Operations on Functions, Kinds of Functions
Quadraticfunctionpresentation 100127142417-phpapp02
Quadraticfunctionpresentation 100127142417-phpapp02
Recently uploaded
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
The typical problem in product engineering is not bad strategy, so much as “no strategy”. This leads to confusion, lack of motivation, and incoherent action. The next time you look for a strategy and find an empty space, instead of waiting for it to be filled, I will show you how to fill it in yourself. If you’re wrong, it forces a correction. If you’re right, it helps create focus. I’ll share how I’ve approached this in the past, both what works and lessons for what didn’t work so well.
ScyllaDB Tablets: Rethinking Replication
ScyllaDB is making a major architecture shift. We’re moving from vNode replication to tablets – fragments of tables that are distributed independently, enabling dynamic data distribution and extreme elasticity. In this keynote, ScyllaDB co-founder and CTO Avi Kivity explains the reason for this shift, provides a look at the implementation and roadmap, and shares how this shift benefits ScyllaDB users.
Freshworks Rethinks NoSQL for Rapid Scaling & Cost-Efficiency
Freshworks creates AI-boosted business software that helps employees work more efficiently and effectively. Managing data across multiple RDBMS and NoSQL databases was already a challenge at their current scale. To prepare for 10X growth, they knew it was time to rethink their database strategy. Learn how they architected a solution that would simplify scaling while keeping costs under control.
"NATO Hackathon Winner: AI-Powered Drug Search", Taras Kloba
This is a session that details how PostgreSQL's features and Azure AI Services can be effectively used to significantly enhance the search functionality in any application.
In this session, we'll share insights on how we used PostgreSQL to facilitate precise searches across multiple fields in our mobile application. The techniques include using LIKE and ILIKE operators and integrating a trigram-based search to handle potential misspellings, thereby increasing the search accuracy.
We'll also discuss how the azure_ai extension on PostgreSQL databases in Azure and Azure AI Services were utilized to create vectors from user input, a feature beneficial when users wish to find specific items based on text prompts. While our application's case study involves a drug search, the techniques and principles shared in this session can be adapted to improve search functionality in a wide range of applications. Join us to learn how PostgreSQL and Azure AI can be harnessed to enhance your application's search capability.
Must Know Postgres Extension for DBA and Developer during Migration
Mydbops Opensource Database Meetup 16
Topic: Must-Know PostgreSQL Extensions for Developers and DBAs During Migration
Speaker: Deepak Mahto, Founder of DataCloudGaze Consulting
Date & Time: 8th June | 10 AM - 1 PM IST
Venue: Bangalore International Centre, Bangalore
Abstract: Discover how PostgreSQL extensions can be your secret weapon! This talk explores how key extensions enhance database capabilities and streamline the migration process for users moving from other relational databases like Oracle.
Key Takeaways:
* Learn about crucial extensions like oracle_fdw, pgtt, and pg_audit that ease migration complexities.
* Gain valuable strategies for implementing these extensions in PostgreSQL to achieve license freedom.
* Discover how these key extensions can empower both developers and DBAs during the migration process.
* Don't miss this chance to gain practical knowledge from an industry expert and stay updated on the latest open-source database trends.
Mydbops Managed Services specializes in taking the pain out of database management while optimizing performance. Since 2015, we have been providing top-notch support and assistance for the top three open-source databases: MySQL, MongoDB, and PostgreSQL.
Our team offers a wide range of services, including assistance, support, consulting, 24/7 operations, and expertise in all relevant technologies. We help organizations improve their database's performance, scalability, efficiency, and availability.
Contact us: info@mydbops.com
Visit: https://www.mydbops.com/
Follow us on LinkedIn: https://in.linkedin.com/company/mydbops
For more details and updates, please follow up the below links.
Meetup Page : https://www.meetup.com/mydbops-databa...
Twitter: https://twitter.com/mydbopsofficial
Blogs: https://www.mydbops.com/blog/
Facebook(Meta): https://www.facebook.com/mydbops/
Demystifying Knowledge Management through Storytelling
The Department of Veteran Affairs (VA) invited Taylor Paschal, Knowledge & Information Management Consultant at Enterprise Knowledge, to speak at a Knowledge Management Lunch and Learn hosted on June 12, 2024. All Office of Administration staff were invited to attend and received professional development credit for participating in the voluntary event.
The objectives of the Lunch and Learn presentation were to:
- Review what KM ‘is’ and ‘isn’t’
- Understand the value of KM and the benefits of engaging
- Define and reflect on your “what’s in it for me?”
- Share actionable ways you can participate in Knowledge - - Capture & Transfer
"Choosing proper type of scaling", Olena Syrota
Imagine an IoT processing system that is already quite mature and production-ready and for which client coverage is growing and scaling and performance aspects are life and death questions. The system has Redis, MongoDB, and stream processing based on ksqldb. In this talk, firstly, we will analyze scaling approaches and then select the proper ones for our system.
Session 1 - Intro to Robotic Process Automation.pdf
👉 Check out our full 'Africa Series - Automation Student Developers (EN)' page to register for the full program:
https://bit.ly/Automation_Student_Kickstart
In this session, we shall introduce you to the world of automation, the UiPath Platform, and guide you on how to install and setup UiPath Studio on your Windows PC.
📕 Detailed agenda:
What is RPA? Benefits of RPA?
RPA Applications
The UiPath End-to-End Automation Platform
UiPath Studio CE Installation and Setup
💻 Extra training through UiPath Academy:
Introduction to Automation
UiPath Business Automation Platform
Explore automation development with UiPath Studio
👉 Register here for our upcoming Session 2 on June 20: Introduction to UiPath Studio Fundamentals: https://community.uipath.com/events/details/uipath-lagos-presents-session-2-introduction-to-uipath-studio-fundamentals/
Northern Engraving | Modern Metal Trim, Nameplates and Appliance Panels
What began over 115 years ago as a supplier of precision gauges to the automotive industry has evolved into being an industry leader in the manufacture of product branding, automotive cockpit trim and decorative appliance trim. Value-added services include in-house Design, Engineering, Program Management, Test Lab and Tool Shops.
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
Mutation Testing for Task-Oriented Chatbots
Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
Dandelion Hashtable: beyond billion requests per second on a commodity server
This slide deck presents DLHT, a concurrent in-memory hashtable. Despite efforts to optimize hashtables, that go as far as sacrificing core functionality, state-of-the-art designs still incur multiple memory accesses per request and block request processing in three cases. First, most hashtables block while waiting for data to be retrieved from memory. Second, open-addressing designs, which represent the current state-of-the-art, either cannot free index slots on deletes or must block all requests to do so. Third, index resizes block every request until all objects are copied to the new index. Defying folklore wisdom, DLHT forgoes open-addressing and adopts a fully-featured and memory-aware closed-addressing design based on bounded cache-line-chaining. This design offers lock-free index operations and deletes that free slots instantly, (2) completes most requests with a single memory access, (3) utilizes software prefetching to hide memory latencies, and (4) employs a novel non-blocking and parallel resizing. In a commodity server and a memory-resident workload, DLHT surpasses 1.6B requests per second and provides 3.5x (12x) the throughput of the state-of-the-art closed-addressing (open-addressing) resizable hashtable on Gets (Deletes).
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
Gursev Pirge, PhD
Senior Data Scientist - JohnSnowLabs
High performance Serverless Java on AWS- GoTo Amsterdam 2024
Java is for many years one of the most popular programming languages, but it used to have hard times in the Serverless community. Java is known for its high cold start times and high memory footprint, comparing to other programming languages like Node.js and Python. In this talk I'll look at the general best practices and techniques we can use to decrease memory consumption, cold start times for Java Serverless development on AWS including GraalVM (Native Image) and AWS own offering SnapStart based on Firecracker microVM snapshot and restore and CRaC (Coordinated Restore at Checkpoint) runtime hooks. I'll also provide a lot of benchmarking on Lambda functions trying out various deployment package sizes, Lambda memory settings, Java compilation options and HTTP (a)synchronous clients and measure their impact on cold and warm start times.
Crafting Excellence: A Comprehensive Guide to iOS Mobile App Development Serv...
Crafting Excellence: A Comprehensive Guide to iOS Mobile App Development Serv...Pitangent Analytics & Technology Solutions Pvt. Ltd
Discover top-tier mobile app development services, offering innovative solutions for iOS and Android. Enhance your business with custom, user-friendly mobile applications.Biomedical Knowledge Graphs for Data Scientists and Bioinformaticians
Dmitrii Kamaev, PhD
Senior Product Owner - QIAGEN
Day 2 - Intro to UiPath Studio Fundamentals
In our second session, we shall learn all about the main features and fundamentals of UiPath Studio that enable us to use the building blocks for any automation project.
📕 Detailed agenda:
Variables and Datatypes
Workflow Layouts
Arguments
Control Flows and Loops
Conditional Statements
💻 Extra training through UiPath Academy:
Variables, Constants, and Arguments in Studio
Control Flow in Studio
From Natural Language to Structured Solr Queries using LLMs
This talk draws on experimentation to enable AI applications with Solr. One important use case is to use AI for better accessibility and discoverability of the data: while User eXperience techniques, lexical search improvements, and data harmonization can take organizations to a good level of accessibility, a structural (or “cognitive” gap) remains between the data user needs and the data producer constraints.
That is where AI – and most importantly, Natural Language Processing and Large Language Model techniques – could make a difference. This natural language, conversational engine could facilitate access and usage of the data leveraging the semantics of any data source.
The objective of the presentation is to propose a technical approach and a way forward to achieve this goal.
The key concept is to enable users to express their search queries in natural language, which the LLM then enriches, interprets, and translates into structured queries based on the Solr index’s metadata.
This approach leverages the LLM’s ability to understand the nuances of natural language and the structure of documents within Apache Solr.
The LLM acts as an intermediary agent, offering a transparent experience to users automatically and potentially uncovering relevant documents that conventional search methods might overlook. The presentation will include the results of this experimental work, lessons learned, best practices, and the scope of future work that should improve the approach and make it production-ready.
Recently uploaded (20)
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
Freshworks Rethinks NoSQL for Rapid Scaling & Cost-Efficiency
Freshworks Rethinks NoSQL for Rapid Scaling & Cost-Efficiency
"NATO Hackathon Winner: AI-Powered Drug Search", Taras Kloba
"NATO Hackathon Winner: AI-Powered Drug Search", Taras Kloba
Must Know Postgres Extension for DBA and Developer during Migration
Must Know Postgres Extension for DBA and Developer during Migration
Demystifying Knowledge Management through Storytelling
Demystifying Knowledge Management through Storytelling
Session 1 - Intro to Robotic Process Automation.pdf
Session 1 - Intro to Robotic Process Automation.pdf
Northern Engraving | Modern Metal Trim, Nameplates and Appliance Panels
Northern Engraving | Modern Metal Trim, Nameplates and Appliance Panels
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
Dandelion Hashtable: beyond billion requests per second on a commodity server
Dandelion Hashtable: beyond billion requests per second on a commodity server
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
High performance Serverless Java on AWS- GoTo Amsterdam 2024
High performance Serverless Java on AWS- GoTo Amsterdam 2024
Crafting Excellence: A Comprehensive Guide to iOS Mobile App Development Serv...
Crafting Excellence: A Comprehensive Guide to iOS Mobile App Development Serv...
Biomedical Knowledge Graphs for Data Scientists and Bioinformaticians
Biomedical Knowledge Graphs for Data Scientists and Bioinformaticians
From Natural Language to Structured Solr Queries using LLMs
From Natural Language to Structured Solr Queries using LLMs
Exponential functions
- 1. Exponential Functions and Their Graphs Digital Lesson
- 2. The exponential function f with base a is defined by f(x) = ax where a > 0, a ≠ 1, and x is any real number. For instance, f(x) = 3x and g(x) = 0.5x are exponential functions.
- 3. The value of f(x) = 3x when x = 2 is f(2) = 32 = The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = The value of f(x) = 3x when x = –2 is 9 1 9 f(–2) = 3–2 = 0.0625
- 4. The Graph of f(x) = ax , a > 1 y x (0, 1) Domain: (–∞, ∞) Range: (0, ∞) Horizontal Asymptote y = 0
- 5. The Graph of f(x) = ax , 0 < a <1 y x (0, 1) Domain: (–∞, ∞) Range: (0, ∞) Horizontal Asymptote y = 0
- 6. Example: Sketch the graph of f(x) = 2x . x x f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 ½ (-1, ½) 0 1 (0, 1) 1 2 (1, 2) 2 4 (2, 4) y 2–2 2 4
- 7. Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. x y The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . f(x) = 2x y = –1 Domain: (–∞, ∞) Range: (–1, ∞) 2 4
- 8. Example: Sketch the graph of g(x) = 2-x . State the domain and range. x y The graph of this function is a reflection the graph of f(x) = 2x in the y- axis. f(x) = 2x Domain: (–∞, ∞) Range: (0, ∞) 2–2 4
- 9. The irrational number e, where e ≈ 2.718281828… is used in applications involving growth and decay. Using techniques of calculus, it can be shown that ∞→→ + ne n n as 1 1
- 10. The Graph of f(x) = ex y x 2–2 2 4 6 x f(x) -2 0.14 -1 0.38 0 1 1 2.72 2 7.39