Chapter 9.2, Problem 46HP

To determine

To describe: The effect of $a\mathrm{and}k$ on the graphical and tabular representations for the functions.

Answer to Problem 46HP

For $y=a{x}^2$ ,the value of a vertical stretches the graph if $\left|a\right|>1$ ,vertically compress it if $0<\left|a\right|<1$ ,and reflect across the $x-\mathrm{axis}$ if $a<0$ .For the tabular representation,the values of $y$ obtained from $y=a{x}^2$ for a couple value of $x$ .

For $y=x^2+k$ ,the value of $k$ translated the graph up $k$ units when $k>0$ and down $\left|k\right|$ units when $k<0$ . For the tabular representation, $k$ is added to the values of $y$ obtained from $y=x^2$ .For example, the tabular representation of $y=x^2-2$ for a couple values of $x$ .

For $y=a{x}^2+k$ ,the value of a vertical stretches the graph if $\left|a\right|>1$ ,vertically compress it if $0<\left|a\right|<1$ ,and reflect across the $x-\mathrm{axis}$ if $a<0$ ,the value of $k$ translated the graph up $k$ units when $k>0$ and down $\left|k\right|$ units when $k<0$ . For the tabular representation the values of $y$ obtained from $y=x^2$ are multiplied by the value of $a$ and then added to the $k$ .For example, the tabular representation of $y=3x^2-2$ for a couple values of $x$ would be

Explanation of Solution

Given information: The given functions is $y=a{x}^2,y=x^2+k\mathrm{and}y=a{x}^2+k$ .

Calculation: The given functions is $y=a{x}^2,y=x^2+k\mathrm{and}y=a{x}^2+k$ .

For $y=a{x}^2$ ,the value of a vertical stretches the graph if $\left|a\right|>1$ ,vertically compress it if $0<\left|a\right|<1$ ,and reflect across the $x-\mathrm{axis}$ if $a<0$ .For the tabular representation,the values of $y$ obtained from $y=a{x}^2$ for a couple value of $x$ would be:

$x$	$x^2$	$y=2x^2$
$0$	$0^2=0$	$2(0)=0$
$1$	$1^2=1$	$2(1)=2$

For $y=x^2+k$ , the value of $k$ translated the graph up $k$ units when $k>0$ and down $\left|k\right|$ units when $k<0$ . For the tabular representation, $k$ is added to the values of $y$ obtained from $y=x^2$ .For example, the tabular representation of $y=x^2-2$ for a couple values of $x$ would be:

$x$	$x^2$	$y=x^2-2$
$0$	$0^2=0$	$0-2=-2$
$1$	$1^2=1$	$1-2=-1$

For $y=a{x}^2+k$ ,the value of a vertical stretches the graph if $\left|a\right|>1$ ,vertically compress it if $0<\left|a\right|<1$ ,and reflect across the $x-\mathrm{axis}$ if $a<0$ ,the value of $k$ translated the graph up $k$ units when $k>0$ and down $\left|k\right|$ units when $k<0$ . For the tabular representation the values of $y$ obtained from $y=x^2$ are multiplied by the value of $a$ and then added to the $k$ .For example, the tabular representation of $y=3x^2-2$ for a couple values of $x$ would be:

$x$	$x^2$	$y=3x^2-2$
$0$	$0^2=0$	$3{(0)}^2-2=0-2=-2$
$1$	$1^2=1$	$3{(1)}^2-2=3-2=1$