Chapter 4.1, Problem 33E

Solution

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

The graph will translate $3$ units to the right and $1$

Given information:

Given that the exponential growth function is $y=a{b}^{x-h}+k$ ,where $a=2,b=5,h=-4,k=3$

Calculation:

The function after substituting the given values becomes:

  $y=2{\left(5\right)}^{x+4}+3$

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 $y=2{\left(5\right)}^x$ by graphing calculator.

The graph of the function $y=2{\left(5\right)}^{x+4}+3$ can be obtained as $h=-4$ the graph will translate $4$ units to the left and $3$ units up.

The graph of both function is given below:

The graph of $y=2{\left(5\right)}^x$ is given below:

  

Now when $a=1,b=4,h=3,k=-1$ the function becomes:

  $y=1{\left(4\right)}^{x-3}-1$

The main function will become $y=4^x$ .

Now the graph will translate $3$ units to the right and $1$

Unit down vertically. The $y$ intercept will be $(0,-1)$ .

The graph of both function given below: