The document defines key terms related to functions, including domain, range, and linear functions. It provides an example of writing a function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The example function is f(c) = 2c + 5, where c is the independent variable representing cups and s is the dependent variable for spoonfuls. A table of values and graph of this linear function are included.
The document defines key vocabulary terms related to functions, including function, domain, range, linear function, and vertical line test. It provides an example of writing a linear function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The function is defined as s=2c+5, and a table is made to graph the relationship between c and s.
This document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the ratio in whole integers. An example demonstrates taking a ratio of 3/4 to 2/3 and rewriting it as 9:8 using cross multiplication. The document also notes that cross multiplication can be used to compare two fractions, with the fraction corresponding to the larger product being the larger fraction.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
The document discusses ratios and proportions. It defines a ratio as two related quantities stated side by side, and gives an example of a 3:4 ratio of eggs to flour in a recipe. It explains how to write ratios as fractions and set up proportion equations. Proportions are equal ratios, like 3:4 being proportional to 6:8. The document solves sample proportion word problems, like finding the number of eggs needed given 10 cups of flour using a proportion equation.
The document discusses proportions and ratios. It defines a ratio as two related quantities stated side by side. It provides an example of a recipe ratio of 3 eggs to 4 cups of flour as 3:4. It explains how to set up proportional equations from word problems by ensuring quantities of the same type occupy the same position in fractions. It solves examples, finding x eggs needed given 10 cups of flour is 7.5 eggs. It also solves a map ratio problem of 4 inches on a map equaling 21 miles in real distance.
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
The document discusses misconceptions about teaching math in the 21st century. The first misconception is that technology needs bells and whistles like animations and distractions, when the focus should be on whether the technology enhances the educational content. The second misconception is that math instruction needs to be entertaining rather than focusing on developing understanding. Effective technology use and making the content meaningful are more important goals for teaching math.
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
The document defines key vocabulary terms related to functions, including function, domain, range, linear function, and vertical line test. It provides an example of writing a linear function to represent the relationship between the number of cups of coffee made and the total number of spoonfuls needed. The function is defined as s=2c+5, and a table is made to graph the relationship between c and s.
This document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the ratio in whole integers. An example demonstrates taking a ratio of 3/4 to 2/3 and rewriting it as 9:8 using cross multiplication. The document also notes that cross multiplication can be used to compare two fractions, with the fraction corresponding to the larger product being the larger fraction.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
The document discusses ratios and proportions. It defines a ratio as two related quantities stated side by side, and gives an example of a 3:4 ratio of eggs to flour in a recipe. It explains how to write ratios as fractions and set up proportion equations. Proportions are equal ratios, like 3:4 being proportional to 6:8. The document solves sample proportion word problems, like finding the number of eggs needed given 10 cups of flour using a proportion equation.
The document discusses proportions and ratios. It defines a ratio as two related quantities stated side by side. It provides an example of a recipe ratio of 3 eggs to 4 cups of flour as 3:4. It explains how to set up proportional equations from word problems by ensuring quantities of the same type occupy the same position in fractions. It solves examples, finding x eggs needed given 10 cups of flour is 7.5 eggs. It also solves a map ratio problem of 4 inches on a map equaling 21 miles in real distance.
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
The document discusses misconceptions about teaching math in the 21st century. The first misconception is that technology needs bells and whistles like animations and distractions, when the focus should be on whether the technology enhances the educational content. The second misconception is that math instruction needs to be entertaining rather than focusing on developing understanding. Effective technology use and making the content meaningful are more important goals for teaching math.
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
This document discusses solving polynomial equations. It begins by working through examples of solving quadratic equations. It then introduces concepts like the Fundamental Theorem of Algebra, which states that every polynomial equation has at least one complex number solution. It discusses double roots and the multiplicity of roots. Finally, it works through examples of determining the number of real solutions a polynomial equation would have based on its degree.
Simulations are used to relate probabilities by modeling complex situations as simple experiments. They involve running trials to represent the complex situation. Examples given include using coins, cards, dice or spinners. The document discusses using the TI-83/84 calculator's ProbSim application to simulate batting averages over multiple at bats and determine the probability of getting a certain number of hits. It provides an example of simulating 10 at bats for someone with a .250 batting average to find the probability of getting exactly 3 hits. Homework assigned is problems 1-10 and odd problems 11-25 on page 155.
The document discusses that there are at least 12 permutations of the letters A, E, P, R, and S that are valid words in Scrabble. These 5 letters form more valid Scrabble words than any other set of 5 letters. It then lists 12 valid permutations of those 5 letters.
This document discusses division of polynomials and the remainder theorem in three sections. It begins with two warm-up problems dividing polynomials and checking the answers by substituting values for x. It then shows the division of 4/12x worked out with coefficients and evaluates the quotient polynomial at x=5, obtaining the expected remainder of 0.
This document provides examples and explanations of trigonometric functions including sine, cosine, and tangent. It discusses how to find the quadrant an image point lies in after a rotation about the origin. It also shows how to use trig functions to find the coordinates of a point on the unit circle after a rotation, and examples of evaluating trig functions with various angle measures. Finally, it gives an example of using trig functions to find the heights of the hour and minute hands of a clock at a certain time.
The document provides examples for writing and graphing linear inequalities in two variables. It defines key vocabulary like open and closed half-planes, boundary lines, and test points. It then works through examples of determining if points satisfy given inequalities and graphing inequalities on a coordinate plane by plotting boundary lines and shading the appropriate half-plane.
The document discusses the rational-zero theorem and examples of applying it to find rational roots of polynomials. Specifically, it states that if p/q is a rational root of a polynomial with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. Two examples are worked through finding possible rational roots by considering factors of the constant and leading coefficients.
The document provides examples of modeling data with polynomials. Example 1 shows data of distance traveled by a ball down an inclined plane over time. A quadratic polynomial model is found to fit this data, with the formula d = 3t^2, where d is distance and t is time. Example 2 shows additional data over time and x values, to be fitted with a polynomial model.
Here are the key steps to solving proportions:
1. Write the proportion as a fraction equal sign.
2. Isolate the variable by multiplying/dividing both sides by the same value.
3. Solve for the variable.
4. Check your solution by plugging it back into the original proportion.
Practice these steps on your homework problems. Let me know if you have any other questions!
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
This document discusses inequalities in one triangle. It defines inequalities and lists properties of inequalities for real numbers. It presents theorems about exterior angle inequality and angle-side relationships in triangles. Examples demonstrate applying these concepts to identify angles and sides of triangles based on given information. The document provides practice problems for students to check their understanding.
The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
This document discusses simplifying variable expressions. It begins with an essential question about adding, subtracting, multiplying, and dividing variable expressions. It then provides the vocabulary for order of operations, including grouping symbols, exponents, multiplication, division, addition, and subtraction. Examples are provided to demonstrate simplifying expressions using order of operations. The examples include solving for variables and writing expressions in terms of variables.
This document discusses solving polynomial equations. It begins by working through examples of solving quadratic equations. It then introduces concepts like the Fundamental Theorem of Algebra, which states that every polynomial equation has at least one complex number solution. It discusses double roots and the multiplicity of roots. Finally, it works through examples of determining the number of real solutions a polynomial equation would have based on its degree.
Simulations are used to relate probabilities by modeling complex situations as simple experiments. They involve running trials to represent the complex situation. Examples given include using coins, cards, dice or spinners. The document discusses using the TI-83/84 calculator's ProbSim application to simulate batting averages over multiple at bats and determine the probability of getting a certain number of hits. It provides an example of simulating 10 at bats for someone with a .250 batting average to find the probability of getting exactly 3 hits. Homework assigned is problems 1-10 and odd problems 11-25 on page 155.
The document discusses that there are at least 12 permutations of the letters A, E, P, R, and S that are valid words in Scrabble. These 5 letters form more valid Scrabble words than any other set of 5 letters. It then lists 12 valid permutations of those 5 letters.
This document discusses division of polynomials and the remainder theorem in three sections. It begins with two warm-up problems dividing polynomials and checking the answers by substituting values for x. It then shows the division of 4/12x worked out with coefficients and evaluates the quotient polynomial at x=5, obtaining the expected remainder of 0.
This document provides examples and explanations of trigonometric functions including sine, cosine, and tangent. It discusses how to find the quadrant an image point lies in after a rotation about the origin. It also shows how to use trig functions to find the coordinates of a point on the unit circle after a rotation, and examples of evaluating trig functions with various angle measures. Finally, it gives an example of using trig functions to find the heights of the hour and minute hands of a clock at a certain time.
The document provides examples for writing and graphing linear inequalities in two variables. It defines key vocabulary like open and closed half-planes, boundary lines, and test points. It then works through examples of determining if points satisfy given inequalities and graphing inequalities on a coordinate plane by plotting boundary lines and shading the appropriate half-plane.
The document discusses the rational-zero theorem and examples of applying it to find rational roots of polynomials. Specifically, it states that if p/q is a rational root of a polynomial with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient. Two examples are worked through finding possible rational roots by considering factors of the constant and leading coefficients.
The document provides examples of modeling data with polynomials. Example 1 shows data of distance traveled by a ball down an inclined plane over time. A quadratic polynomial model is found to fit this data, with the formula d = 3t^2, where d is distance and t is time. Example 2 shows additional data over time and x values, to be fitted with a polynomial model.
Here are the key steps to solving proportions:
1. Write the proportion as a fraction equal sign.
2. Isolate the variable by multiplying/dividing both sides by the same value.
3. Solve for the variable.
4. Check your solution by plugging it back into the original proportion.
Practice these steps on your homework problems. Let me know if you have any other questions!
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
This document discusses inequalities in one triangle. It defines inequalities and lists properties of inequalities for real numbers. It presents theorems about exterior angle inequality and angle-side relationships in triangles. Examples demonstrate applying these concepts to identify angles and sides of triangles based on given information. The document provides practice problems for students to check their understanding.
The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
The document defines various terms related to angle measure including ray, angle, vertex, acute angle, obtuse angle, and angle bisector. It then provides examples measuring angles in a figure and solving an equation involving angle measures.
This document discusses finding points and midpoints on line segments. It defines midpoint as the point halfway between two endpoints and provides the formula to calculate it. Several examples are given to demonstrate how to find the midpoint of a segment, locate a point at a fractional distance from one endpoint, and find a point where the ratio of distances from the endpoints is a given ratio. The key concepts covered are calculating midpoints using averages of x- and y-coordinates, and setting up and solving equations to locate interior points using fractional or ratio distances along a segment.
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
This document introduces basic geometry concepts such as points, lines, planes, and their intersections. It defines a point as having no size or shape, a line as an infinite set of collinear points, and a plane as a flat surface extending indefinitely. Examples demonstrate identifying geometric shapes from real-world objects and graphing points and lines on a coordinate plane. The summary defines key terms and provides examples of geometric concepts and relationships.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
The document discusses composition of functions. It defines composition of functions as using the output of one function as the input of another. It provides an example of composing two functions f and g, showing the steps of evaluating f(g(x)) and g(f(x)) at different values of x. Another example is given with two functions defined by sets of ordered pairs, finding the compositions f∘g and g∘f by evaluating them at different inputs and stating their domains and ranges.
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
The document discusses determining the number and type of roots of polynomial equations. It states that every polynomial with degree greater than zero has at least one root in the set of complex numbers according to the Fundamental Theorem of Algebra. Descartes' Rule of Signs is introduced, which relates the number of changes in sign of a polynomial's terms to its possible positive and negative real roots. An example problem is worked through applying these concepts to determine the possible number of positive, negative, and imaginary roots.
This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.
This document discusses solving polynomial equations by factoring polynomials. It begins with essential questions and vocabulary about factoring polynomials and solving polynomial equations by factoring. It then provides the number of terms in a polynomial and the corresponding factoring technique that can be used. Examples of factoring various polynomials are also provided. The document aims to teach students how to factor polynomials and solve polynomial equations by factoring.
The document defines key terms and theorems related to trapezoids and kites. It provides definitions for trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, and kite. It also lists theorems about properties of isosceles trapezoids and kites. Two examples problems are included, one finding measures of an isosceles trapezoid and another showing a quadrilateral is a trapezoid.
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
The document discusses properties of rectangles. A rectangle is defined as a parallelogram with four right angles. The key properties are that opposite sides are parallel and congruent, opposite angles are congruent, and consecutive angles are supplementary. The diagonals of a rectangle bisect each other and are congruent. Theorems are presented regarding the diagonals of rectangles. Examples apply the properties of rectangles to find missing side lengths, angles, and diagonals. One example uses the distance formula and slope to determine if a quadrilateral is a rectangle.
The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
The document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with two pairs of parallel sides. It then lists several properties of parallelograms: opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, and if one angle is a right angle all angles are right angles. It also discusses properties of diagonals in parallelograms, including that diagonals bisect each other and divide the parallelogram into two congruent triangles. Several examples demonstrate using these properties to solve problems about parallelograms.
The document summarizes key concepts about polygons, including:
- The sum of the interior angles of a polygon with n sides is (n-2)180 degrees.
- The sum of the exterior angles of a polygon is 360 degrees.
- Examples are provided to demonstrate calculating sums of interior/exterior angles and finding missing angle measures using angle sums.
- Regular polygons are defined by their number of sides.
The document discusses analyzing graphs of polynomial functions. It provides examples of locating real zeros of polynomials using the location principle and estimating relative maxima and minima. Example 1 analyzes the polynomial f(x) = x^4 - x^3 - 4x^2 + 1 and locates its real zeros between consecutive integer values. Example 2 graphs the polynomial f(x) = x^3 - 3x^2 + 5 and estimates the x-coordinates of relative maxima and minima.
This document discusses polynomial functions. It defines key terms like polynomial in one variable, leading coefficient, and polynomial function. It provides examples of power functions of varying degrees like quadratic, cubic, quartic and quintic functions. The document also includes examples of evaluating polynomial functions, finding degrees and leading coefficients, graphing polynomial functions from tables of values, and describing properties of graphs.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
2. Essential Questions
How do you graph linear and nonlinear functions?
How do you identify the domain and range of a function?
Where you’ll see this:
Machinery, travel, temperature
4. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation:
3. Domain:
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:
5. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain:
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:
6. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range:
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:
7. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range: Any possible value for the dependent variable (usually y)
5. Continuous:
6. Linear Function:
7. Vertical-Line Test:
8. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range: Any possible value for the dependent variable (usually y)
5. Continuous: A graph where all points are connected
6. Linear Function:
7. Vertical-Line Test:
9. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range: Any possible value for the dependent variable (usually y)
5. Continuous: A graph where all points are connected
6. Linear Function: A function that will give a straight line; any line other
than a vertical line
7. Vertical-Line Test:
10. Vocabulary
1. Function: A relationship where each x-value (independent variable)
matches with only one y-value (dependent variable)
2. Function Notation: f(x), reads “function of x”; tells us the independent
variable is inside the parentheses; allows for working with multiple
functions
3. Domain: Any possible value for the independent variable (usually x)
4. Range: Any possible value for the dependent variable (usually y)
5. Continuous: A graph where all points are connected
6. Linear Function: A function that will give a straight line; any line other
than a vertical line
7. Vertical-Line Test: Tests whether a graph represents a function or not;
can only touch a graph once
11. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
12. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
13. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
14. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) =
15. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) = 2c
16. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) = 2c + 5
17. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) = 2c + 5
s = 2c + 5
18. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) = 2c + 5 Function
s = 2c + 5
19. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
a. Write a function where c is the number of cups of coffee being
made and s is the total number of spoonfuls of coffee used.
Which is the independent variable?
The number of cups determines how many spoonfuls, so c is independent
f(c) = 2c + 5 Function
s = 2c + 5 Equation
20. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
21. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c
22. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
23. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1
24. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7
25. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
26. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2
27. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9
28. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
29. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3
30. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11
31. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
32. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4
33. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13
34. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
35. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5
36. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15
37. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
38. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
39. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
40. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
41. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
42. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
43. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
44. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
45. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
46. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
47. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
b. Make a table and graph the data.
c s s
1 7 (1, 7)
2 9 (2, 9)
3 11 (3, 11)
4 13 (4, 13)
5 15 (5, 15)
c
48. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
c. An office cafeteria has a coffee urn with the ability to make 16 to 35
cups. Determine the domain and range of the function as applied to
this urn.
49. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
c. An office cafeteria has a coffee urn with the ability to make 16 to 35
cups. Determine the domain and range of the function as applied to
this urn.
Domain: {c : c = 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35}
50. Example 1
To make coffee in a large coffee urn, one recipe calls for two
spoonfuls for each cup plus 5 spoonfuls for the pot.
c. An office cafeteria has a coffee urn with the ability to make 16 to 35
cups. Determine the domain and range of the function as applied to
this urn.
Domain: {c : c = 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35}
Range: {s : s = 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65,
67, 69, 71, 73, 75}
51. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
52. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
53. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x
54. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
55. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2
56. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1
57. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
58. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1
59. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0
60. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
61. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0
62. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1
63. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
64. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1
65. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2
66. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
67. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2
68. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3
69. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
70. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
71. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
72. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
73. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
74. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
75. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
76. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
x y
-2 1 (-2, 1)
-1 0 (-1, 0)
0 1 (0, 1)
1 2 (1, 2)
2 3 (2, 3)
77. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
78. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain:
79. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}
80. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}
Range:
81. Example 2
Graph the following relation, stating whether it is a function and
listing the domain and range.
y = x +1
Domain: {x : x is all real numbers}
Range: {y : y ≥ 0}
82. Homework
p. 267 #1-23 odd
"That is what learning is.You suddenly understand something you've
understood all your life, but in a new way." - Doris Lessing