Chapter 8.4, Problem 108E

To determine

The polynomial function g in standard form.

Answer to Problem 108E

  $g\left(x\right)=-x^4+12x^3-50x^2+84x-46$

Explanation of Solution

Given:

Two functions are as

  $\begin{array}{l}f\left(x\right)=–x^4+4x^2–1\hfill \\ g\left(x\right)=f\left(x–3\right)\hfill \end{array}$

By applying transformations on $f(x)$ let's convert it to $g(x)$ .

Replace f(x) by $f\left(x-3\right)$ .

According to Binomial Theorem Expansion of ${(x+y)}^n$ is given as

  $x^n+n{x}^{n-1}y+\cdots {+}_n{C}_r{x}^{n-r}{y}^r+\cdots +nx{y}^{n-1}+y^n$

  $\begin{array}{c}$ g(x)=-{(x-3)}^4+4{(x-3)}^2-1$\begin{array}{l}=-\left(x^4-4x^3(3)+6x^2{(3)}^2-4x{(3)}^3+81\right)+4\left(x^2-6x+9\right)-1\hfill \\ =-\left(x^4-12x^3+54x^2-108x+81\right)+\left(4x^2-24x+36\right)-1\hfill \\ =-x^4+12x^3-50x^2+84x-46\hfill \end{array}\end{array}$

Obtain the graph of the above functionsas,

  

The graph is similar in structure just apart from each other by 3 units.