Chapter ISG, Problem 7.6.3P

To determine

To graph: The transformed graphs of the function $y=5\cdot 3^{x+2}-4$ and $y=-3\cdot 2^{-x}+1$ form their parent function.

Explanation of Solution

Given information: The graph of the function $y=3^x$ is given below. Its transformed function is $y=5\cdot 3^{x+2}-4$ .

  

Figure (1)

The graph of the function $y=2^x$ is given below. Its transformed function is $y=-3\cdot 2^{-x}+1$ .

  

Figure (2)

Graph:

The transformation of graph of exponential function from parent function to other equation can be drawn by the general equation:

  $y=a\cdot b^{x-h}+k$ …… (1)

Here, $h$ shows horizontal shift, $k$ shows vertical shift and $a$ defines stretch or compression.

Consider the function $y=5\cdot 3^{x+2}-4$ . The graph of its parent function $y=3^x$ is given. Compare the transformation function with the equation (1) to get,

  $\begin{array}{l}a=5\\ h=-2\\ k=-4\end{array}$

To draw the graph of the function $y=5\cdot 3^{x+2}-4$ , first shift the graph of its parent function 2 units to the left, stretch vertical as $a>1$ , and then shift 5 units downwards as given below.

  

Figure (3)

Consider the function $y=-3\cdot 2^{-x}+1$ . The graph of its parent function $y=2^x$ is given. Compare the transformation function with the equation (1) to get,

  $\begin{array}{l}a=-5\\ h=0\\ k=1\end{array}$

Negative power of exponent shows the mirror image of the graph.

To draw the graph of the function $y=-3\cdot 2^{-x}+1$ , first draw the mirror image of the graph of its parent, shift the graph 3 of the function 2 units to the left and make a reflection across $x-$ axis and stretch vertical as $a>1$ . Now, shift the graph 1 unit upwards as given below.

  

Figure (4)

Interpretation: The graph of both the functions is drawn by the graph of their parent function. Graph explains the horizontal shift, vertical shift, reflection and stretch or compression of the function.