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This document discusses absolute value functions and their graphs. It defines the parent graph of an absolute value function as f(x)=x and describes its V-shape and axis of symmetry. Examples are given of absolute value functions after various transformations from the parent graph, along with the general form y=a(x-h)+k. The document also describes how to identify the vertex and axis of symmetry from the equation of an absolute value function and how to write the equation of an absolute value function given its graph. Homework problems are assigned from the textbook.
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Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
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This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
2.6
There are families of functions that share characteristics, with a parent function being the simplest form. Transformations of the parent function include translations, which shift the graph horizontally and/or vertically without changing shape; reflections, which flip the graph over an axis; and vertical stretches or compressions, which multiply or reduce the y-values by a factor. Combining transformations results in a new function, and the document provides examples of determining rules for functions related through transformations.
Math 10.1
1. Write the function in standard form y=ax^2+bx+c.
2. Find the vertex by using the formula x=-b/2a.
3. Find the axis of symmetry by setting x=-b/2a.
4. Sketch the parabola using the vertex and axis of symmetry as guides.
Math 10.1
1. Write the function in standard form y=ax^2+bx+c.
2. Find the vertex by using the formula x=-b/2a.
3. Find the axis of symmetry by setting x=-b/2a.
4. Sketch the parabola using the vertex and axis of symmetry, opening up or down depending on whether a is positive or negative.
Alg II 2-7 Transformations
This document provides an overview of key concepts for graphing and understanding absolute value functions in Algebra II Chapter 2. It defines absolute value functions and their key features, including that the absolute value of f(x) gives the distance from the y-axis. Students will learn to graph absolute value functions by hand and using technology. The general form of an absolute value function is given as y = a|x - h| + k and examples are provided to show transformations from the standard form. Practice problems are assigned from the textbook.
Lecture Notes In Algebra
- A rational function is a function that can be written as the ratio of two polynomial functions. Graphs of rational functions may have breaks in continuity where there is a vertical asymptote or point discontinuity.
- Mathematician Maria Gaetana Agnesi discussed the "curve of Agnesi", the equation x2/y = a2(a - y), which can be expressed as a rational function y = a2/x2 + a2.
- A polynomial is the sum of monomial terms. The degree of a polynomial is the degree of the highest-degree term. The FOIL method can be used to multiply binomials by distributing and combining like terms.
Chapter 4 and half
- A rational function is a function that can be written as the ratio of two polynomial functions. Graphs of rational functions may have breaks in continuity where there is a vertical asymptote or point of discontinuity.
- Mathematician Maria Gaetana Agnesi discussed the "curve of Agnesi", the equation x2/y = a2(a - y), which can be written as a rational function y = a2/x2 + a2.
- A polynomial is the sum of monomial terms. The degree of a polynomial is the degree of the highest-degree term. The FOIL method can be used to multiply binomials by distributing and combining like terms.
Quadratic functions
The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the
G9_Q2_W2_L2_GraphsofQUadraticFunction.pdf
The document discusses key concepts about quadratic functions including:
1) Quadratic functions can be represented by the equation f(x) = ax^2 + bx + c and form a parabolic shape that can open up or down.
2) The axis of symmetry of a parabola passes through the vertex (highest/lowest point) and can be found using the equation x = -b/2a.
3) The vertex coordinates can be determined by substituting the axis of symmetry value into the original equation.
4) Quadratic functions can have 0, 1, or 2 x-intercepts depending on the number of real solutions to the related quadratic equation.
Unit v
The document discusses various mathematical methods including matrices, eigen values, quadratic forms, algebraic and transcendental equations, interpolation, curve fitting, numerical differentiation and integration, Fourier series and transforms, and partial differential equations. It provides the contents, recommended textbooks and references for further study on these topics which are important for engineering applications.
vvvvvvvvvvvvvL2A_CurveRepresentations.pdf
This document discusses different representations of curves and surfaces in computer graphics, including explicit, implicit, and parametric representations. Explicit representations express one variable as a function of others, but cannot represent all shapes. Implicit representations use equations like f(x,y)=0 but can be difficult to render points on. Parametric representations express each coordinate as a function of one or more parameters and have advantages over explicit and implicit forms like versatility and ease of rendering. The document focuses on using rational parametric polynomials, as they are simple yet powerful representations.
Quadraticfuntions
A quadratic function has the form f(x) = ax^2 + bx + c, where a != 0. The graph of a quadratic function is a parabola, which is symmetric around its axis of symmetry. Any quadratic function can be written in standard form by completing the square, where the standard form is f(x) = a(x-h)^2 + k. The vertex of the parabola is the point (h, k) and the line of symmetry is x = h.
Quadraticfuntions
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a is not equal to zero. The graph of a quadratic function is a parabola, which is symmetric around its axis of symmetry. Any quadratic function can be written in standard form by completing the square, where the standard form is f(x) = a(x-h)^2 + k. The vertex of the parabola is the point (h, k) and finding the standard form makes it easy to identify the vertex.
Chapter on Functions and Graphs.ppt
A relation is a set of ordered pairs. A function is a relation where each domain value has only one range value. To determine if a relation is a function, use the vertical line test or check if each x-value only has one y-value. An equation defines a function if each x only corresponds to one y when solving the equation for y. Piecewise functions are defined by two or more equations over different parts of the domain. The slope of a line is rise over run and can be found by calculating change in y over change in x between any two points. You can write the equation of a line from its point-slope form or by finding the slope between two points and plugging into point-slope form
Quadratics
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form - and how to graph each. It provides the key characteristics and steps to graph a quadratic function for each form. Standard form is ax2 + bx + c, vertex form is a(x-h)2 + k, and intercept form is a(x-p)(x-q). The document explains how to identify the vertex, axis of symmetry, intercepts, and whether the graph opens up or down based on the coefficients for each form. It also gives the process to follow to plot points and sketch the parabolic graph.
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7.3 daqy 2
Logarithmic functions are inverses of exponential functions. To graph a logarithmic function:
1. Identify the inverse exponential form.
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7.3
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The document discusses exponential functions of the form f(x) = a*b^x, explaining that they always have a curved shape and asymptote at y=0. It distinguishes between exponential growth, where the value of y increases as x increases, and exponential decay, where the value of y decreases as x increases. Examples are provided to demonstrate how to determine if a function represents growth or decay and to find the y-intercept.
7.2
This document discusses properties and transformations of exponential functions, including stretch, compression, reflection, and horizontal and vertical translation. It also discusses the number e as the base for natural exponential functions and using the function A=Pe^rt to model continuously compounded interest. Examples are provided to demonstrate graphing transformations of exponential functions and using the continuously compounded interest formula.
10.1 part2
This document discusses testing for homogeneity among populations using chi-square tests. It defines homogeneity as populations having the same structure or composition. A test of homogeneity determines if different populations have the same proportions for various categories. It requires using a contingency table and chi-square distribution. An example tests if the same proportion of males and females prefer different pet types using survey data from college students.
10.1 part 1
The document provides an overview of the chi-square distribution and how it can be used for hypothesis testing. It discusses that the chi-square distribution is used to find critical values for determining the area under the curve for a given degrees of freedom. It also gives an example of how chi-square can be used to test if two variables such as keyboard type and time to learn typing are independent.
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This document discusses synthetic division and the remainder theorem. Synthetic division is a process that simplifies long division when dividing a polynomial by a linear factor of the form x - a. It involves setting up the coefficients of the polynomial and multiplying/adding through the process. The remainder theorem states that if a polynomial P(x) is divided by x - a, then the remainder is equal to P(a). It provides a quick way to find the remainder of a polynomial division problem by evaluating the polynomial at the value of a. Examples are given to demonstrate evaluating polynomials using the remainder theorem.
5.4 long division
Long division can be used to divide polynomials in a similar way to dividing numbers. The key steps are to set up the division problem, divide the term of the dividend by the term of the divisor, multiply the divisor by the quotient term and subtract, then bring down the next term of the dividend and repeat. This polynomial long division allows polynomials to be factored by finding all divisor polynomials that give a remainder of zero. The factor theorem can also be used to check if a linear polynomial is a factor by setting it equal to zero and checking if it makes the other polynomial equal to zero.
5.3
This document discusses solving polynomial equations by factoring. It provides examples of factoring polynomials, including factoring the difference and sum of cubes. Factoring by substitution is also introduced as a method for factoring polynomials of degree 4 or higher. The document demonstrates solving polynomial equations by factoring the expressions and setting each factor equal to 0. Both real and imaginary solutions may be obtained depending on whether the factors are real or complex numbers. Graphing is presented as an alternative method to find real solutions of a polynomial equation.
9.3 Part 1
The document discusses inferences for correlation and regression. It provides an example of testing the correlation between percentage of population with a college degree (x) and percentage growth in income (y) for 6 Ohio communities. There is a positive correlation between x and y, but this does not necessarily mean higher education causes higher earnings. The document also discusses measuring the spread of data points around the least squares line, including the standard error of estimate, using an example of how much copper sulfate dissolves in water at different temperatures.
5.2
1. The document discusses writing polynomials in factored form and finding the zeros of polynomial functions. It defines linear factors, roots, zeros, and x-intercepts as equivalent terms.
2. Examples are provided of writing polynomials in factored form using the factor theorem to find the zeros, and then graphing the polynomial function based on its zeros.
3. The factor theorem states that a linear expression x - a is a factor of a polynomial if and only if a is a zero of the related polynomial function. This allows writing a polynomial given its zeros.
5.1 part 2
This document discusses how to describe the shape of a cubic function by listing it in standard form, describing the end behavior of the graph, determining the possible number of turning points using a table of values, and determining the increasing and decreasing intervals. It explains that to describe the shape, you identify the sign of the leading coefficient to determine the end behavior and the number of turning points, which is one less than the possible degree. The document also discusses using differences of consecutive y-values in a table to determine the least degree of the polynomial function that could generate the data, with constant first differences indicating linear, constant second differences indicating quadratic, and constant third differences indicating cubic.
5.1[1]
This document defines key concepts related to polynomials and polynomial functions. It defines monomials as terms involving variables and exponents, and polynomials as sums of monomials. The degree of a polynomial is the highest exponent among its terms. Polynomial functions are polynomials written in terms of a single variable. Standard form arranges polynomial terms by descending degree. Polynomials are classified by degree and number of terms. Higher degree polynomials can have more turning points and their end behavior depends on the leading term. Examples show determining standard form, classifying polynomials, identifying end behavior and increasing/decreasing parts of graphs.
9.1
This document discusses scatter diagrams and linear correlation. It provides examples of scatter diagrams that do and do not show linear correlation. It defines the correlation coefficient r as a measure of linear correlation between two variables on a scatter plot, with values between -1 and 1. It presents formulas for calculating r and provides an example of computing r using wind velocity and sand drift rate data. It cautions that correlation does not necessarily imply causation and that lurking variables can influence the correlation between two variables.
9.2 lin reg coeff of det
1) Linear regression finds the "best-fitting" linear relationship between two variables by minimizing the vertical distances between the data points and the linear equation line.
2) The coefficient of determination, r^2, measures how well the linear relationship described by the regression line fits the actual data, with higher r^2 values indicating less unexplained variability.
3) r^2 has an interpretation as the percentage of the total variation in the response variable that is explained by the explanatory variable.
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2.7
- 1. 2.7 ABSOLUTE VALUE FUNCTIONS AND GRAPHS
- 2. ABSOLUTE VALUE FUNCTION ï‚¢ The simplest form (parent graph) of an absolute value function is f ( x ) = x
- 3. ABSOLUTE VALUE FUNCTION ï‚¢ The graph of f ( x ) = x is V-shaped and symmetric about a vertical line called the axis of symmetry. ï‚— The graph also has a single maxima point or minima point, called a vertex
- 4. EXAMPLE Make a table of values, then graph the function. Describe the transformation from the parent function. y = x −4
- 5. EXAMPLE Make a table of values, then graph the function. Describe the transformation from the parent function. f ( x) = x + 2 + 3
- 6. EXAMPLE Graph the equation. Describe the transformation from the parent function. 1 y= x 2
- 7. EXAMPLE Graph the equation. Describe the transformation from the parent function.
- 8. TRANSFORMATIONS OF THE ABSOLUTE VALUE FUNCTION
- 9. GENERAL FORM OF THE ABSOLUTE VALUE FUNCTION When translations, stretches and compressions are combined for Absolute Value Functions, we derive the general form: y =a x−h +k The stretch or compression factor is |a|, the vertex is located at (h, k), and the axis of symmetry is the line x = h
- 10. EXAMPLE Without graphing, what are the vertex and axis of symmetry of the graph of: y =3 x−2 +4
- 11. EXAMPLE Without graphing, what are the vertex and axis of symmetry of the graph of: y = −2 x − 1 − 3
- 12. EXAMPLE Without graphing, what are the vertex and axis of symmetry of the graph of:
- 13. WRITING AN ABSOLUTE VALUE FUNCTION FROM A GRAPH 1. Identify the vertex 2. Identify a, by finding the slope of the branch on the right 3. Write the equation
- 14. EXAMPLE What is the equation of the absolute value function?
- 15. EXAMPLE What is the equation of the absolute value function?
- 16. WITHOUT USING A TABLE OF VALUES GRAPH THE FUNCTION
- 17. HOMEWORK  Page 111  #9 – 21 odd  #23 – 30 all