Chapter 7.5, Problem 13WE

To determine

Draw the given equations on the same axis.

Explanation of Solution

Given Information:

Given equations are :

  $\begin{array}{l}y=x^2\\ y={\left(x-2\right)}^2\\ y={\left(x+3\right)}^2\end{array}$

Calculation:

• Parent Function: $y=x^2$
• Vertex = (0,0)
• Type of Graph:

Upward Parabola

• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

  $x=0$

• In the equation $y={\left(x-2\right)}^2$ ; comparing it with standard form of parabola equation $Y=X^2$

$y=Y;{\left(x-2\right)}^2=X^2$

• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y=Y=0;\left(x-2\right)=X=0\\ \Rightarrow y=0;x=2\\ \Rightarrow Vertex\equiv (2,0)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=X}}^2]\\ \Rightarrow \left(x-2\right)=\mathrm{0........}[X=\left(x-2\right)]\\ \Rightarrow x=\mathrm{2..................}[\text{add 2 each side]}\end{array}$

The equation of axis of symmetry is x = 2

• In the equation $y={\left(x+3\right)}^2$ ; comparing it with standard form of parabola equation $Y=X^2$

  $y=Y;{\left(x+3\right)}^2=X^2$

• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y=Y=0;\left(x+3\right)=X=0\\ \Rightarrow y=0;x=-3\\ \Rightarrow Vertex\equiv (-3,0)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=X}}^2]\\ \Rightarrow \left(x+3\right)=\mathrm{0........}[X=\left(x+3\right)]\\ \Rightarrow x=-\mathrm{3..................}[\text{subtract 3 from each side]}\end{array}$

The equation of axis of symmetry is x = -3

Use the above information to graph the equations

  $\begin{array}{l}y=x^2\\ y={\left(x-2\right)}^2\\ y={\left(x+3\right)}^2\end{array}$: