MATH 110 College Algebra Online Families of Functions Transformations Functions are important in mathematics. Being able to tell what family a function comes from, its domain and range and finding a function s inverse if it has one, are all important parts of College Algebra. Lesson 11.1 discusses three families of functions the linear function, the absolute value function and the quadratic function. Thoroughly review this lesson, making sure you can identify the graph of the function, find the domain and range, and find the x- and y-intercepts. Also, for the absolute value and quadratic functions you should be able to find the vertex. Finally, study the application problem presented on pages 5 and 6. The purpose of this worksheet is to practice taking a basic function and transforming it on a graph. A transformation is when a graph has been moved up or down, right or left, flipped across an axis, or stretched and shrunk. A summary of this procedure for an absolute value function is on page 16 of your text and discussed through examples on pages 15 and 16. Examples for the quadratic function are given on pages 19 and 0. The following examples illustrate Vertical shifts (moving up or down), Horizontal shifts (moving left or right), Reflections about the x-axis (flipping a graph over the x-axis) and Stretching/Shrinking a graph. When more than one of these transformations occurs on a graph, they are done in the order of Horizontal Shift, Stretch/Shrink, Reflect, Vertical Shift or HSRV. A student of mine named AJ coined the phrase, I live in a HouSe and drive an RV, to help her remember the order. The following are examples of Vertical shifts of the basic graph f ( x) 1. f ( x) = x + 3 (shifts 3 units up). f ( x) x = (shifts units down) Notice that the shape of the function did not change when moving the graph up and down. C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 1
The following examples are Horizontal shifts of f ( x) 3. f ( x) = x + 3 (shifts 3 units left) 4. f ( x) x = (shifts units right) Notice the shape of the function did not change when moving the graph left and right. Also notice in number 3, when 3 was added to x the graph moved left, in number 4, when was subtracted from x the graph moved right. The following is an example of a Reflection of f ( x) 5. f ( x) = x Notice the shape did not change when the graph was reflected (flipped) about the x-axis. The negative sign in front of the function does the reflection. Finally, here are examples of Stretching/Shrinking the graph of f ( x) 6. f ( x) = 3 x (stretches away from the x-axis) 7. ( ). f x = 05 x (shrinks toward the x-axis) Notice that the shape of the functions DID change when the basic function was multiplied by a number. C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page
Now let s use more than one transformation of f ( x) 8. f ( x) = x + 4 1 (Remember HSRV!) Horizontal shift: f ( x) = x + 4 1 Move graph of f ( x) = x left 4 units Stretch/Shrink: ( ) 4 1 f x = x + Stretch the graph away from the x-axis Reflect: f ( x) = x + 4 1 Flip the graph over the x-axis Vertical shift: f ( x) = x + 4 1 Move the graph down one unit C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 3
Now let s use the function f ( x) = x to practice transformations. 1. f ( x) = ( x + 1). f ( x) x = + 3 Horizontal shift one unit left Vertical shift 3 units up 3. f ( x) x 4. f ( x) = 4 x Reflection about x-axis Stretch away from x-axis f x = x 3 5 5. f ( x) = 0. x 6. ( ) ( ) Shrink towards the x-axis 3-R -S 1-H 4-V HSRV: 1-H move graph 3 units right -S stretch graph away from x-axis 3-R flip the graph over the x-axis 4-V move the graph down 5 units C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 4
MATH 110 College Algebra Online Families of Functions Transformations Name Date Problems 1-3: Graph the following. 1. f ( x) = x. f ( x) = x 3. f ( x) = x C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 5
Problems 4-13: Graph the following by first sketching the basic function and then performing one transformation at a time. Darken your final answer so that it stands out. 4. f ( x) = x + 3 5. ( ) ( ) f x = x + 1 6. f ( x) x = + 7. f ( x) = 3 x + 1 4 C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 6
f x = 4 x 3 + 8. ( ) ( ) 9. f ( x) x + 1 + 3 f x = x + 1 + 3 10. ( ) ( ) 11. Start with f ( x) = x and write the equation that would transform the graph of f ( x ) 3 units right, stretch it away from the x-axis by a factor of and drop it down units. 1. Start with f ( x) = x and write the equation that would transform the graph of f ( x ) units left, shrink it by a factor of 0., and reflect it across the x-axis. C:\Documents and Settings\fdc1.DOMAIN1\Desktop\MA 110 Families of Functions and Transformations Sum 07.doc Page 7