Example 1

Manipulate the animation in figure 3 to answer the following questions:

1. What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[/latex] when it is transformed to [latex]f(x)=4\left(2^x\right)[/latex]?

2. What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[/latex] when it is transformed to [latex]f(x)=4\left(3^x\right)[/latex]?

3. What happens to the point (0, 1) on the graph of [latex]f(x)=\left(\dfrac{1}{2}\right)^x[/latex] when it is transformed to [latex]f(x)=4\left(\dfrac{1}{2}\right)^x[/latex]?

4. What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=4r^x[/latex]?

5. What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=ar^x[/latex]?

Solution

1. With [latex]r=2[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).

2. With [latex]r=3[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).

3. With [latex]r=\dfrac{1}{2}[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).

4. With [latex]a=4[/latex], the point (0, 1) moves to [latex](0, 4)[/latex].

5. The point (0, 1) moves to [latex](0, a)[/latex].

Example 2

Determine what happens to the point (2, 9) when the function [latex]f(x)=3^x[/latex] is:

1. stretched by a factor of 5

2. compressed to [latex]\dfrac{1}{3}[/latex] its height

Solution

1. When a graph is stretched the [latex]x[/latex]-coordinate stays the same, while the [latex]y[/latex]-coordinate is multiplied by the stretch factor, 5. Hence, (2, 9) becomes (2, 45).

2. When a graph is compressed the [latex]x[/latex]-coordinate stays the same, while the [latex]y[/latex]-coordinate is multiplied by[latex]\dfrac{1}{3}[/latex]. Hence, (2, 9) becomes (2, 3).

Example 3

The function [latex]f(x)=4^x[/latex] is transformed. Determine the equation of the transformed function when:

1. [latex]f(x)[/latex] is stretched by a factor of 2.

2. [latex]f(x)[/latex] is stretched by a factor of 5.

3. [latex]f(x)[/latex] is compressed to [latex]\dfrac{1}{3}[/latex] its height.

4. [latex]f(x)[/latex] is compressed to [latex]\dfrac{1}{4}[/latex] its height.

Solution

1. Since [latex]a=2[/latex] the transformed function is [latex]f(x)=2\left(4^x\right)[/latex].

2. Since [latex]a=5[/latex] the transformed function is [latex]f(x)=5\left(4^x\right)[/latex].

3. Since [latex]a=\dfrac{1}{3}[/latex] the transformed function is [latex]f(x)=\dfrac{1}{3}\left(4^x\right)[/latex].

4. Since [latex]a=\dfrac{1}{4}[/latex] the transformed function is [latex]f(x)=\dfrac{1}{4}\left(4^x\right)[/latex].

Example 4

The function [latex]f(x)=4^x[/latex] is reflected across the [latex]x[/latex]-axis. What happens to the point (2, 16) that lies on the parent function after the transformation?

Solution

When a function is reflected across the [latex]x[/latex]-axis, the [latex]x[/latex]-coordinate stays the same while the [latex]y[/latex]-coordinate changes sign. So (2, 16) is transformed to (2, –16).

Example 5

The function [latex]f(x)=4^x[/latex] is reflected across the [latex]y[/latex]-axis. What happens to the point (2, 16) that lies on the parent function, after the transformation?

Solution

When a function is reflected across the [latex]y[/latex]-axis, the [latex]y[/latex]-coordinate stays the same while the [latex]x[/latex]-coordinate changes sign. So (2, 16) is transformed to (–2, 16).

Example 6

What transformations were performed on the parent function [latex]f(x)=2^x[/latex] to get the function [latex]f(x)=-3\left(2^{x+4}\right)-6[/latex]?

Solution

First we identify [latex]a, h,[/latex] and [latex]k[/latex]:

[latex]a=-3,\;h=-4,\;k=-6[/latex]

A negative value of [latex]a[/latex] means the function [latex]f(x)=2^x[/latex] has been reflected across the [latex]x[/latex]-axis. [latex]a=-3[/latex] means it has also been stretched by a factor of 3.

[latex]h=-4[/latex] means the parent function has been shifted horizontally 4 units left, while [latex]k=-6[/latex] means it has been shifted vertically down by 6 units.

Example 7

If the parent function [latex]g(x)=3^x[/latex] is stretched by a factor of 7, reflected across the [latex]y[/latex]-axis, and shifted vertically 3 units down, what is the equation of the transformed function?

Solution

Stretched by a factor of 7 means [latex]a=7[/latex].

Reflected across the [latex]y[/latex]-axis means [latex]x[/latex] becomes [latex]-x[/latex].

Shifted vertically down by 3 units means [latex]k=-3[/latex].

So, the transformed function is [latex]f(x)=7\left(3^{-x}\right)-3[/latex].