Chapter ISG, Problem 7.6.2P

To determine

To explain: The changes in the shape of the graph of an exponential function if function is multiplied by a number between 0 and $-1$ , and if exponent is multiplied by a number between 0 and $-1$ .

Answer to Problem 7.6.2P

If an exponential function is multiplied by a number between 0 and $-1$ , then it forms a reflection across $x-$ axis and compress towards $x-$ axis, and if exponent of the function is multiplied by a number between 0 and $-1$ , then it forms a mirror image becomes wider from $y-$ axis.

Explanation of Solution

Given information: The exponential function and exponent should be multiplied by a number between 0 and $-1$ .

Calculation:

Consider an exponential function.

  $f\left(x\right)=2^x$

Multiply the exponential function by $-\frac{1}{2}$ between 0 and $-1$ .

  $f\left(x\right)=-\frac{1}{2}\left(2^x\right)$

Multiply the exponent by $-\frac{1}{2}$ between 0 and $-1$ .

  $f\left(x\right)=2^{-\frac{1}{2}x}$

Draw the graph of all three functions in the same viewing window as shown below:

  

From the above graph it can be observed that if an exponential function is multiplied by a number between 0 and $-1$ , then it forms a reflection across $x-$ axis and compress towards $x-$ axis.

If exponent of the function is multiplied by a number between 0 and $-1$ , then it forms a mirror image becomes wider from $y-$ axis.

Therefore, if an exponential function is multiplied by a number between 0 and $-1$ , then it forms a reflection across $x-$ axis and compress towards $x-$ axis, and if exponent of the function is multiplied by a number between 0 and $-1$ , then it forms a mirror image becomes wider from $y-$ axis.