Chapter 2.2, Problem 37E

To determine

To analyze: The function which has the same axis of symmetry.

Answer to Problem 37E

Option B $y=-3x^2-6x+2$ has the same axis of symmetry as the graph of $y=x^2+2x+2$ .

Explanation of Solution

Given information: $y=x^2+2x+2$ with four different functions.

Axis of symmetry of $y=x^2+2x+2$, General form of a quadratic function $y=a{x}^2+bx+c$ ,

From the given function $y=x^2+2x+2$ ,

  $a=1,b=2,c=2$

  $x-$ coordinate of the vertex is $x=-\frac{b}{2a}$ .

  $\begin{array}{l}x=-\frac{2}{2\times 1}\\ x=-\frac{2}{2}\\ x=-1\end{array}$

The axis of symmetry of the function $y=x^2+2x+2$ is $x=-1$ .

Axis of symmetry of option A, $y=2x^2+2x+2$

General form of a quadratic function $y=a{x}^2+bx+c$ ,

From the given function $y=2x^2+2x+2$ ,

  $a=2,b=2,c=2$

  $x-$ coordinate of the vertex is $x=-\frac{b}{2a}$ .

  $\begin{array}{l}x=-\frac{2}{2\times 2}\\ x=-\frac{2}{4}\\ x=-\frac{1}{2}\\ x\ne -1\end{array}$

The axis of symmetry of the function of option A $y=2x^2+2x+2$ is not same as $y=x^2+2x+2$ .

Axis of symmetry of option B, $y=-3x^2-6x+2$

General form of a quadratic function $y=a{x}^2+bx+c$ ,

From the given function $y=-3x^2-6x+2$ ,

  $a=-3,b=-6,c=2$

  $x-$ coordinate of the vertex is $x=-\frac{b}{2a}$ .

  $\begin{array}{l}x=-\frac{-6}{2\times (-3)}\\ x=-\frac{-6}{-6}\\ x=-1\end{array}$

The axis of symmetry of the function of option B $y=-3x^2-6x+2$ is same as $y=x^2+2x+2$ .

Axis of symmetry of option C, $y=x^2-2x+2$

General form of a quadratic function $y=a{x}^2+bx+c$ ,

From the given function $y=x^2-2x+2$ ,

  $a=1,b=-2,c=2$

  $x-$ coordinate of the vertex is $x=-\frac{b}{2a}$ .

  $\begin{array}{l}x=-\frac{-2}{2\times (1)}\\ x=-\frac{-2}{2}\\ x=1\\ x\ne -1\end{array}$

The axis of symmetry of the function of option C $y=x^2-2x+2$ is not same as $y=x^2+2x+2$ .

Axis of symmetry of option D, $y=-5x^2+10x+2$

General form of a quadratic function $y=a{x}^2+bx+c$ ,

From the given function $y=-5x^2+10x+2$ ,

  $a=-5,b=10,c=2$

  $x-$ coordinate of the vertex is $x=-\frac{b}{2a}$ .

  $\begin{array}{l}x=-\frac{10}{2\times (-5)}\\ x=-\frac{10}{-10}\\ x=1\\ x\ne -1\end{array}$

The axis of symmetry of the function of option D $y=-5x^2+10x+2$ is not same as $y=x^2+2x+2$ .

Thus we can conclude that option B $y=-3x^2-6x+2$ has the same axis symmetry as the graph $y=x^2+2x+2$ .