Chapter 4.7, Problem 24E

To determine

To write: a rule for $g$ to describe the indicated transformation of graph of $f$ .

Answer to Problem 24E

  $g\left(x\right)=2x^4-32x^3+192x^2-508x+508$

Explanation of Solution

Given information:

Given function is,

  $f\left(x\right)=x^4+2x+6$

The indicated transformation is

Vertical stretch by a factor of 2, followed by a translation 4 units right.

Concept used:

The graph of $g\left(x\right)=f\left(x-h\right)$ can be found by translating the graph of $f\left(x\right)$ by $h$ units right horizontally.

The graph of $g\left(x\right)=af\left(x\right)$ can be found by stretching the graph of $f\left(x\right)$ by a factor of $a$ units vertically.

Calculation:

Consider the given function.

  $f\left(x\right)=x^4+2x+6$

Apply the transformation “Vertical stretch by a factor of 2”.

  $\begin{array}{c}2f\left(x\right)=2\left(x^4+2x+6\right)\\ =2x^4+4x+12\end{array}$

Now, apply the transformation “translation 4 units right” as shown:

  $\begin{array}{c}g\left(x\right)=f\left(x-2\right)\\ =2{\left(x-2\right)}^4+4\left(x-2\right)+12\\ =2x^4-16x^3+32x^2-16x^3+128x^2-256x+32x^2-256x+512+4x-16+12\\ =2x^4-32x^3+192x^2-508x+508\end{array}$