Chapter 7.5, Problem 11WE

To determine

Draw the graph .Find the vertex , axis of symmetry and all the intercepts.

Answer to Problem 11WE

Vertex : $(-1,-8)$

Axis of Symmetry : x = -1

y -intercept : (0,-7.5)

x -intercepts are (-5,0) and (3,0)

Explanation of Solution

Given Information:

Given equation is :

  $y+8=\frac{1}{2}{\left(x+1\right)}^2$

Concept Used :

• 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.

  $x=0$

• y -intercept = The y coordinate of the point at which the graph intersects y -axis , x = 0.
• x -intercept = The x coordinate of the point at which the graph intersects x -axis , y = 0 .

Calculation:

In the equation $y+8=\frac{1}{2}{\left(x+1\right)}^2$ ; comparing it with standard form of parabola equation $Y=X^2$, opens upward , has vertex (0,0) , axis of symmetry is X = 0.

$y+8=Y;\frac{1}{2}{\left(x+1\right)}^2={\left(\frac{1}{\sqrt{2}}\left(x+1\right)\right)}^2=X^2$

Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y+8=Y=0;\frac{1}{\sqrt{2}}\left(x+1\right)=X=0\\ \Rightarrow y=-8;x=-1\\ \Rightarrow Vertex\equiv (-1,-8)\end{array}$

• Find Axis of Symmetry :

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

The equation of axis of symmetry is x = -1

• Find y -intercept :

The y -coordinate of the point at which the graph cuts the y -axis is the y -intercept.

Hence , substituting x =0 in the equation $y+8=\frac{1}{2}{\left(x+1\right)}^2$ , we will get the y -intercept.

  $\begin{array}{l}y+8=\frac{1}{2}{\left(x+1\right)}^2\\ \Rightarrow y+8=\frac{1}{2}{\left(0+1\right)}^2\mathrm{.............}[x=0]\\ \Rightarrow y+8=\frac{1}{2}{\left(1\right)}^2\\ \Rightarrow y=\frac{1}{2}-\mathrm{8.........................}[subtract\text{ 8 from each side]}\\ \Rightarrow y=-7.5\end{array}$

So, y -intercept = (0,-7.5)

• Find x -intercept :

The x -coordinate of the point at which the graph cuts the x -axis is the x -intercept.

Hence , substituting y =0 in the equation $y+8=\frac{1}{2}{\left(x+1\right)}^2$ , we will get the x -intercept.

  $\begin{array}{l}y+8=\frac{1}{2}{\left(x+1\right)}^2\\ \Rightarrow 0+8=\frac{1}{2}{\left(x+1\right)}^2\mathrm{.............}[y=0]\\ \Rightarrow 8=\frac{1}{2}{\left(x+1\right)}^2\\ \Rightarrow {\left(x+1\right)}^2=\mathrm{16....................}[\text{multiply each side by 2]}\\ \Rightarrow x+1=\pm \mathrm{4.......................}[\text{take square root each side]}\\ \Rightarrow x=-1\pm \mathrm{4......................}[\text{subtract 1 from each side]}\\ \Rightarrow x=3\text{ and }x=-5\end{array}$

So, x -intercepts are (3,0) and (-5,0).

Use the above information to graph the equation $y+8=\frac{1}{2}{\left(x+1\right)}^2$: