Chapter 8.4, Problem 107E

To determine

The polynomial function g in standard form. Relation between the graphs of $f(x)$ and $g(x)$ .

Answer to Problem 107E

  $g\left(x\right)=x^4-8x^3+19x^2-12x-4$

The graph is similar in structure just apart from each other by 2 units.

Explanation of Solution

Given:

Two functions are as

  $\begin{array}{l}f\left(x\right)=x^4-5x^2\hfill \\ g\left(x\right)=f\left(x–2\right)\hfill \end{array}$

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

Replace $f(x)$ by $f\left(x-2\right)$ .

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-2)}^4-5{(x-2)}^2$\begin{array}{l}=\left(x^4-4x^3(2)+6x^2{(2)}^2-4x{(2)}^3+16\right)-5\left(x^2-4x+4\right)\hfill \\ =\left(x^4-8x^3+24x^2-32x+16\right)-\left(5x^2-20x+20\right)\hfill \\ =x^4-8x^3+19x^2-12x-4\hfill \end{array}\end{array}$

Obtain the graph of the above functionsas,

  

The graph is similar in structure just apart from each other by 2 units.