Chapter 9.3, Problem 2CCYP

To determine

Describe how the graph of each function is related to the graph of $f(x)=x^2\text{and}g(x)=\frac{1}{3}{x}^2+2$

Answer to Problem 2CCYP

The graph of $g(x)=\frac{1}{3}{x}^2+2$ is the graph of $f(x)=x^2$ vertically compressed by a factor $\frac{1}{3}$ and shifted up 2 units.

Explanation of Solution

Given: The functions $f(x)=x^2\text{and}g(x)=\frac{1}{3}{x}^2+2$

Concept Used:

  $\begin{array}{l}\text{Vertical}\text{Stretch}\text{and}\text{Vertical}\text{Compression:}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{a}\text{>1}\text{,}\text{Srtetch}\text{the}\text{graph}\text{of}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\text{.}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{0}\text{<}\text{a}\text{<1}\text{,}\text{Compress}\text{the}\text{graph}\text{of}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\\ \end{array}$

The function can be written $f(x)=a{x}^2+c$, Where $a=\frac{1}{3}\text{and}c=+2$. When c is positive, it translated up and if c is negative, it translated down. Since $0<a<1$ and c is positive, the graph of $g(x)=\frac{1}{3}{x}^2+2$ is the graph of $f(x)=x^2$ vertically compressed by a factor $\frac{1}{3}$ and shifted up 2 units.

Thus, The graph of $g(x)=\frac{1}{3}{x}^2+2$ is the graph of $f(x)=x^2$ vertically compressed by a factor $\frac{1}{3}$ and shifted up 2 units.