Lesson 4.5

y x

f x x

The graph of the square root function, , is another parent function that you can use to illustrate transformations. From the graphs below, what are the domain and range of ?

f x xGraph on your calculator. You can see that at x=3, f(x)≈ 1.732.

y x

f x x

The graph of the square root function, , is another parent function that you can use to illustrate transformations. From the graphs below, what are the domain and range of ?

f x xWhat is the approximate value of at x = 8?

31How would you use the graph to find ?

What happens when you try to find f (x) for values of x < 0?

In this investigation you first will work with linear functions to discover how to create a new transformation—a reflection. Then you will apply reflections to quadratic functions and square root functions.

1  0.5   2f x x Graph on your calculator.

a. Predict what the graph of -f1(x) will look like. Then check your prediction by graphing f2(x)= -f1(x).

b. Change f1 to f1(x)=-2x-4, and repeat the instructions in part a.

c. Change f1 to f1(x)=x2 +1 and repeat.d. In general, how are the graphs of y=f(x) and

y=-f (x) related?

1  0.5   2f x x Graph on your calculator.

a. Predict what the graph of f1(-x) will look like. Then check your prediction by graphing f2(x)= f1(-x).

b. Change f1 to f1(x)=-2x-4, and repeat the instructions in part a.

c. Change f1 to f1(x)=x2 +1 and repeat.d. Change f1 to f1(x)=(x-3)2+2 and repeat.

Explain what happens.e. In general, how are the graphs of y=f(x) and

y=f (-x) related?

a)Predict what the graphs of f2 =- f1(x) and f3= f1(-x) will look like. Use your calculator to verify your predictions. Write equations for both of these functions in terms of x.

b)Predict what the graph of f4 =-f1(-x) will look like. Use your calculator to verify your prediction.

c) Notice that the graph of the square root function looks like half of a parabola. Why isn’t it an entire parabola? What function would you graph to complete the bottom half of the parabola?

Graph on your calculator.1( )f x x

Reflection of a Function

A reflection is a transformation that flips a graph across a line, creating a mirror image.

Given the graph of y=f(x), • the graph of y=f(-x) is a horizontal

reflection across the y-axis, and• the graph of -y= f(x), or y=-f(x), is a vertical

reflection across the x-axis.

A piecewise function is a function that consists of two or more ordinary functions defined on different domains.

2 , 3 0( )

,0 4

x xf x

x x

Graph

Enter the equation in your graphing calculator.

Find an equation for the piecewise functionpictured at right.

For -4≤x≤0, the function appears to be equal to a reflection of the square root function about the y-axis and then shifted one unit up. This would be the function ( ) 1f x x

For 0<x≤3, the function appears to be equal to a reflection of the square curve reflected over the x-axis and then shifted one unit to the right and 2 units up.

2( ) ( 1) 2f x x

Find an equation for the piecewise functionpictured at right.

2

1 , 4 0

( 1) 2, 0 3

x x

x x