Chapter 6.4, Problem 36E

To determine

To calculate: A rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)={\left(\frac{2}{3}\right)}^x$ , reflection in the x-axis, followed by a vertical stretch by a factor of $6$ and a translation $4$ units left.

Answer to Problem 36E

The rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)={\left(\frac{2}{3}\right)}^x$ is $g\left(x\right)=-6{\left(\frac{2}{3}\right)}^{x+2}$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)={\left(\frac{2}{3}\right)}^x$.

Calculation:

  $f\left(x\right)={\left(\frac{2}{3}\right)}^x$

First, reflect it across the x-axis.

For reflection in x-axis, change the sign of the given function $f\left(x\right)$ that is $-f\left(x\right)$ . Thus, after the translation the function becomes

  $f\left(x\right)=-f\left(x\right)$

  $f\left(x\right)=-{\left(\frac{2}{3}\right)}^x$

After that, vertically stretch the obtained graph by factor of $6$ .

For the stretching, multiply the given function $f\left(x\right)$ with the given unit.

[Hint: For stretching $f\left(x\right)$ by $c$ units, multiply $c$ by the function $f\left(x\right)$ , that is, $cf\left(x\right)$ ]

Therefore, after the reflection the rule of the transformed function $f\left(x\right)$ of the given function $f$ is $f\left(x\right)=-6{\left(\frac{2}{3}\right)}^x$.

And finally, translate it 4 units left.

For translation in left by some units, add $x$ with the given unit.

[Hint: For translating $f\left(x\right)$ by $d$ units set, $f\left(x+d\right)$ ]

Therefore, after translation the function becomes,

  $f\left(x\right)=f\left(x+2\right)$

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=-6{\left(\frac{2}{3}\right)}^{x+2}$ .