Chapter 7.5, Problem 16WE

To determine

Draw the given equations on the same axis.

Explanation of Solution

Given Information:

Given equations are :

  $\begin{array}{l}y=\frac{1}{2}{x}^2\\ y-2=\frac{1}{2}{\left(x-1\right)}^2\end{array}$

Calculation:

• Parent Function: $y=\frac{1}{2}{x}^2$
• 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 $y-2=\frac{1}{2}{\left(x-1\right)}^2$ ; comparing it with standard form of parabola equation $Y=X^2$
• $y-2=Y;{\left(\frac{1}{\sqrt{2}}(x-1)\right)}^2=X^2$

• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y-2=Y=0;\frac{1}{\sqrt{2}}\left(x-1\right)=X=0\\ \Rightarrow y=2;x=1\\ \Rightarrow Vertex\equiv (1,2)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=X}}^2]\\ \Rightarrow \frac{1}{\sqrt{2}}\left(x-1\right)=\mathrm{0........}[X=\frac{1}{\sqrt{2}}\left(x-1\right)]\\ \Rightarrow x=\mathrm{1..................}[\text{multiply each side by }\sqrt{2}\text{ and add 1 each side]}\end{array}$

The equation of axis of symmetry is x = 1

Use the above information to graph the equations

  $\begin{array}{l}y=\frac{1}{2}{x}^2\\ y-2=\frac{1}{2}{\left(x-1\right)}^2\end{array}$: