Chapter 4.2, Problem 25E

Solution

To find: The effect on the function’s graph of each change in $a,b,h,k$ .

The graph will translate $5$ units to the right and $3$ unit up vertically.

Given information:

Given that the exponential growth function is $y=a{b}^{x-h}+k$ ,where $a=3,b=0.4,h=2,k=-1$

Calculation:

The function after substituting the given values becomes:

  $y=3{\left(0.4\right)}^{x-2}-1$

Compare this equation with $y=a{b}^{x-h}+k$ here $a$ is vertical shrink or stretch of the graph, the $y$ intercept is $\left(0,a\right)$ , $h,k$ denotes the translation of graph to right ,left or down and up respectively.

First sketch the graph of $y=3{\left(0.4\right)}^x$ by graphing calculator.

The graph of the function $y=3{\left(0.4\right)}^{x-2}-1$ can be obtained as $h=2$ the graph will translate $2$ units to the right and $1$ units down.

The graph of both function is given below:

  

Now when $a=4,b=0.2,h=5,k=3$ the function becomes:

  $y=4{\left(0.2\right)}^{x-5}+3$

The main function will become $y=4{\left(0.2\right)}^x$ .

Now the graph will translate $5$ units to the right and $3$ unit up vertically.

The graph of both function given below: