Chapter 2.5, Problem 7AYU

In problems 7-18, match each graph to one of the following functions:

A. $y\text{ = }{x}^2\text{ + 2}$     B. $y\text{ = -}{x}^2\text{ + 2}$     C. $y\text{ = }\left|x\right|\text{ + 2}$     D. $y\text{ = -}\left|x\right|\text{ + 2}$     

E. $y\text{ = }{\left(x\text{ - 2}\right)}^2$     F. $y\text{ = -}{\left(x\text{ + 2}\right)}^2$     G. $y\text{ = }\left|x\text{ - 2}\right|$     H. $y\text{ = -}\left|x\text{ + 2}\right|$     

I. $y\text{ = 2}{x}^2$     J. $y\text{ = -2}{x}^2$     K. $y\text{ = 2}\left|x\right|$     L. $y\text{ = -2}\left|x\right|$     

To determine

To find: The function $y$ for the graph.

Answer to Problem 7AYU

$y=-x^2+2$

Explanation of Solution

Given:

Formula Used:

Draw the graph of $ƒ$	Functional change to $ƒ\left(x\right)$	To Graph
Vertical shift up $ƒ$ by $k$ units: Raise the graph of $ƒ$ by $k$ units.	add $k$ to $ƒ\left(x\right)$.	$y=ƒ\left(x\right)+k$
Reflection about the $x\text{-axis}$: Reflect the graph of $ƒ$ about the $x\text{-axis}$.	multiply $ƒ\left(x\right)$ by $-1$.	$y=-ƒ\left(x\right)$

Calculation:

Let the function $ƒ\left(x\right)=x^2$.

Step 1: Reflection about $x\text{-axis}$.

Multiply $x^2$ by $-1=-x^2$.

Step 2: Shift up 2 units.

add 2 to $-x^2=-x^2+2$.

$y=-x^2+2$