Chapter 6.3, Problem 55E

To determine

To describe: the effect of a on the graph of $y=a\cdot 2^x$ when a is positive and when a is negative.

Explanation of Solution

Concept used:

Vertical transformation:

Relative to the graph of $f\left(x\right)$ , the transformation of graph of $g\left(x\right)=cf\left(x\right)$ is a vertical shrink when $0<c<1$ and is a vertical stretch when $c>1$ (where each y value is multiplied by c).

Reflection in coordinate axes:

The graph of $g\left(x\right)=f\left(-x\right)$ is the reflection in the y-axis of the graph of f.

The graph of $g\left(x\right)=-f\left(x\right)$ is the reflection in the x-axis of the graph of f.

Consider the function $y=a\cdot 2^x$ ,

First let a is positive.

Relative to the graph of $y=2^x$ , the transformation of graph of $y=a\cdot 2^x$ is a vertical shrink by a factor of a when $0<a<1$ and is a vertical stretch by a factor of a when $a>1$ .

Now, let a is negative.

Relative to the graph of $y=2^x$ , the transformation of graph of $y=a\cdot 2^x$ is a vertical shrink by a factor of a when $0<a<1$ and is a vertical stretch by a factor of a when $a>1$ . And a reflection in the x-axis of the graph of $y=2^x$.