Chapter 7.6, Problem 75PFA

To determine

To identify the transformed function $g(x)$ which is the translation of the function $f(x)={\frac{1}{2}}^x$ to 1 unit left, stretched vertically by a factor of 7 and reflected across x -axis.

Answer to Problem 75PFA

  $g(x)=-7{\left(\frac{1}{2}\right)}^{x+1}$

Explanation of Solution

Given:

Parent function: $f(x)={\frac{1}{2}}^x$

Given,

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

Now, taking the reflection of the given function about x -axis, which means,

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

Now, stretching the obatined function vertically by factor of 7, which means,

  $y=-7f(x)=-7{\left(\frac{1}{2}\right)}^x$

Now, shift the obtained function 1 unit to the left, which means,

  $y=-7f(x+1)=-7{\left(\frac{1}{2}\right)}^{x+1}$

Thus, the above expression represents the required function: $g(x)=-7f(x+1)=-7{\left(\frac{1}{2}\right)}^{x+1}.$

Conclusion:

Hence, the required function is: $g(x)=-7{\left(\frac{1}{2}\right)}^{x+1}.$