Parent Functions and Transformations PDF

Here is the conversion of the provided text into a structured markdown format: # 1.5 Parent Functions and Transformations **Then** * You analyzed graphs of functions. (Lessons 1-2 through 1-4) **Now** 1. Identify, graph, and describe parent functions. 2. Identify and graph transformations of...

Here is the conversion of the provided text into a structured markdown format: # 1.5 Parent Functions and Transformations **Then** * You analyzed graphs of functions. (Lessons 1-2 through 1-4) **Now** 1. Identify, graph, and describe parent functions. 2. Identify and graph transformations of parent functions. **Why?** The path of a 60-yard punt can be modeled by the function at the right. This function is related to the basic quadratic function $f(x) = x^2$. *Image of the Punted Football Graph* **New Vocabulary** * Parent Function * Constant Function * Zero Function * Identity Function * Quadratic Function * Cubic Function * Square Root Function * Reciprocal Function * Absolute Value Function * Step Function * Greatest Integer Function * Transformation * Translation * Reflection * Dilation ## 1 Parent Functions A family of functions is a group of functions with graphs that display one or more similar characteristics. A **Parent Function** is the simplest of the functions in a family. This is the function that is transformed to create other members in a family of functions. In this lesson, you will study eight of the most commonly used parent functions. You should already be familiar with the graphs of the following linear and polynomial parent functions. **Key Concept: Linear and Polynomial Parent Functions** * **A constant function** has the form $f(x) = c$, where c is any real number. Its graph is a horizontal line. when c = 0, f(x) is the zero function. *Image of Constant Function graph* * **The identity function** $f(x) = x$ passes through all points with coordinates (a, a). *Image of Identity Function graph* * **The quadratic function** $f(x) = x^2$ has a U-shaped graph. *Image of Quadratic Function graph* * **The cubic function** $f(x) = x^3$ is symmetric about the origin. *Image of Cubic Function graph* You should also be familiar with the graphs of both the square root and reciprocal functions. **Key Concept: Square Root and Reciprocal Parent Functions** * The square root function has the farm $f(x) = \sqrt x$. *Image of Square Root Function graph* * The reciprocal function has the form $f(x) = \frac{1}{x}$. *Image of Reciprocal Function graph* Another parent function is the piecewise-defined absolute value function. **Key Concept: Absolute Value Parent Function** | Words | Model | | :----- | :------------------------------------------------------------------------------------------ | | "The absolute value function, denoted $f(x) = \lvert x \rvert $, is a V-shaped function defined as: $f(x) = \begin{cases} -x \quad \text{if } x<0 \\ x \quad \text{if } x\ge0 \end{cases} $ | *Image of absolute value function graph* | | Examples $\lvert -5 \rvert = 5, \lvert 0 \rvert = 0, \lvert 4 \rvert = 4$ | | A piecewise-defined function in which the graph resembles a set of stairs is called a **step function**. The most well-known step function is the greatest integer function **Key Concept: Greatest Integer Parent Function** | Words | Model | | :----- | :------------------------------------------------------------------------------------------ | | The greatest integer function, denoted $f(x) = \lfloor x \rfloor$, is defined as the greatest integer less than or equal to x. | *Image of Greatest Integer Function graph* | | Examples $\lfloor -4 \rfloor=-4, \lfloor-1.5 \rfloor = -2 , \lfloor \frac{3}{2} \rfloor = 1$ | | Using the tools you learned in Lessons 1-1 through 1-4, you can describe characteristics of each parent function. Knowing the characteristics of a parent function can help you analyze the shapes of more complicated graphs in that family. **Example 1 Describe Characteristics of a Parent Function** Describe the following characteristics of the graph of the parent function $f(x) = \sqrt x$: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. The graph of the square root function (*Image of Square Root Function Graph*) has the following characteristics. * The domain of the function is $[0, \infty)$, and the range is $[0, \infty)$. * The graph has one intercept at (0, 0). * The graph has no symmetry. Therefore, f(x) is neither odd nor even. * The graph is continuous for all values in its domain. * The graph begins as x=0 and $\lim_{x \to \infty} f(x) = \infty$. * The graph is increasing on the interval (0, ∞). **Guided Practice** 1. Describe the following characteristics of the graph of the parent function $f(x) = \vert x \vert$: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. ## 2 Transformations **Transformations** of a parent function can affect the appearance of the parent graph. **Rigid transformations** change only the position of the graph, leaving the size and shape unchanged. **Nonrigid transformations** distort the shape of the graph. A **translation** is a rigid transformation that has the effect of shifting the graph of a function. A vertical translation of a function *f* shifts the graph of *f* up or down, while a horizontal translation shifts the graph left or right. Horizontal and vertical translations are examples of rigid transformations. **Key Concept: Vertical and Horizontal Translations** | Vertical Translations | Horizontal Translations | | :------------------------------------------------------------------------------------------ | :-------------------------------------------------------------------------------- | | The graph of $g(x) = f(x) + k$ is the graph of *f(x)* translated: * k units up when *k > 0*, and * k units down when *k < 0*. | The graph of $g(x) = f(x-h)$ is the graph of *f(x)* translated: * h units right when *h > 0*, and * h units left when *h < 0* | | *Vertical Translation Graphs Image* | *Horizontal Translation Graphs Image* | **Example 2 Graph Translations** Use the graph of $f(x) = \lvert x \rvert$ to graph each function. a. $g(x) = \lvert x \rvert + 4$ This function is of the form $g(x) = f(x) + 4$. So, the graph of g(x) is the graph of $f(x)=\lvert x \rvert$ translated 4 units up (*Image of vertical translation up graph*). b. $g(x) = \lvert x + 3 \rvert$ This function is of the form $g(x) = f(x + 3)$ or $g(x) = f(x - (-3))$. So, the graph of g(x) is the graph of $f(x)= \lvert x \rvert$ translated 3 units left (*Image of horizontal translation left graph*). c. $g(x) = \lvert x - 2 \rvert - 1$ This function is of the form $g(x) = f(x - 2) - 1$. So, the graph of g(x) is the graph of $f(x)=\lvert x \rvert$ translated 2 units right and 1 unit down (*Image of combined translation graph*) **Guided Practice** Use the graph of $f(x) = x^3$ to graph each function. 2A. $h(x) = x^3 - 5$ 2B. $h(x) = (x - 3)^3$ 2C. $h(x) = (x + 2)^3 + 4$ Another type of rigid transformation is a **reflection**, which produces a mirror image of the graph of a function with respect to a specific line. **Key Concept: Reflections in the Coordinate Axes** | Reflection in x-axis | Reflection in y-axis | | :------------------------------------------ | :------------------------------------------ | | $g(x) = -f(x)$ is the graph of *f(x)* reflected in the x-axis | $g(x) = f(-x)$ is the graph of *f(x)* reflected in the y-axis | | *Image of x-axis reflection* | *Image of y-axis reflection* | When writing an equation for a transformed function, be careful to indicate the transformations correctly. The graph of $g(x) = -\sqrt{x-1} +2$ is different from the graph of $g(x) = -(\sqrt{x} -1 + 2)$. *Image comparing order of operations in transformations* **Example 3 Write Equations for Transformations** Describe how the graphs of $f(x) = x^2$ and $g(x)$ are related. Then write an equation for g(x). **a.** *Image of Quadratic Translation a* **b.** *Image of Quadratic Translation b* * The graph of g(x) is the graph of f(x) = $x^2$ translated 2.5 units to the right and reflected in the x-axis. So, $g(x) = -(x - 2.5)^2$. * The graph of g(x) is the graph of f(x) = x² reflected in the x-axis and translated 2 units up. So, $g(x) = -x^2 + 2$. **Guided Practice** Describe how the graphs of $f(x) = \frac{1}{x}$ and g(x) are related. Then write an equation for g(x). **3A.** *Image of Hyperbola graph A* **3B.** *Image of Hyperbola graph B* A **dilation** is a nonrigid transformation that has the effect of compressing (shrinking) or expanding (enlarging) the graph of a function vertically or horizontally. **Key Concept: Vertical and Horizontal Dilations** | Vertical Dilations | Horizontal Dilations | | :---------------------------------------------------------------------------------------- | :---------------------------------------------------------------------------------------- | | If a is a positive real number, then g(x) = a * f(x) is: * The graph of *f(x)* expanded vertically, if *a > 1*. * The graph of *f(x)* compressed vertically, if *0 < a < 1*. | If a is a positive real number, then g(x) = f(ax) is: * The graph of *f(x)* compressed horizontally, if *a > 1*. * The graph of *f(x)* expanded horizontally, if *0 < a < 1*. | | *Image of Vertical Dilations graph* | *Image of Horizontal Dilations graph* | **Example 4 Describe and Graph Transformations** Identify the parent function f(x) of g(x), and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes. a. $g(x) = \frac{1}{4}x^3$ *Image of Cubic Dilation a Graph* The graph of g(x) is the graph of $f(x) = x^3$ compressed vertically because $g(x) = \frac{1}{4} x^3 = \frac{1}{4}f(x)$ and $0<\frac{1}{4}<1$. b. $g(x) = -(0.2x)^2$ *Image of Quadratic Dilation b Graph* The graph of g(x) is the graph of $f(x) = x^2$expanded horizontally and then reflected in the x-axis because $g(x) = -(0.2x)^2 = f(0.2x)$ and $0 < 0.2 < 1$ **Guided Practice** 4A. g(x) = [x] - 4 *Image of Greatest Integer Graph A* 4B. $g(x) = \sqrt{x} + 3$ *Image of Greatest Integer Graph B* You can use what you have learned about transformations of functions to graph piecewise-defined function. **Example 5 Graph a Piecewise-Defined Function** Graph $f(x) = \begin{cases} 3x^2 & \text{if } x<-1 \\ -1 & \text{if } -1 \le x < 4 \\ (x-5)^3+2 & \text{if } x\ge 4 \end{cases} $ On the interval (-∞, -1), graph y = $3x^2$. On the interval [-1, 4), graph the constant function y = -1. On the interval [4,∞), graph y = $(x-5)^3+2$. Draw circles at (-1, 3) and (4, -1) and dots at (-1, -1) and (4, 1) because f(-1) = -1 and f(4) = 1 (*Image of piece-wise defined function graph*) **Guided Practice** Graph each function. 5A. $g(x) = \begin{cases} x-5 & \text{if } x \le 0 \\ x^3 & \text{if } 0 2) is a horizontal translation of 10 yards to the right, so, g(x) = 60x+1. **Electriciy:The current in amps flowing through a DVD player is described by I(x)=,where x is the power in watts and r is the resistance in ohms .** **Electriciy:The resistance of lamp is 15 ohms.Write a fuction to describe the current flowing through the lamp.** Let the electric current flowing through the lamp be I(x)=√. **Graph the resistance for the DVD player and the lamp on the same graphing calculator seeen.** All the graphs intersect. Another nonrigid transformation involves absolute value. **Key Concept;Traansformations with Absolute Value** g(x)=lf(x)l This transformation reflects any portion of the graph of f(x) that is the below x axi so that it ist above the x axis. g(x)=fIxI) This transformation results in the portion of the graph of f(x) that is to the left of the y axis being replaced by reflection of the portion to the right of the y axis. (a,b)→(IaI,b) **EXAMPLE 8.Transformations** Use the graph of flx) in Figure 1.5.6 to graph each function. a- g(x):l f(x)l *Image transformationA* b- b(x):fllxl *IMage of trnsformationB*. Guided pratcie 480 **Lessons**