Chapter 7.5, Problem 10WE

To determine

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

Answer to Problem 10WE

Vertex : $(3,8)$

Axis of Symmetry : x = 3

y -intercept : (0,-10)

x -intercepts : (1,0) and (5,0)

Explanation of Solution

Given Information:

Given equation is :

  $y-8=-2{\left(x-3\right)}^2$

Concept Used :

• Parent Function:

  $y=-x^2$

• Vertex = (0,0)
• Type of Graph:

Downward 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 $y-8=-2{\left(x-3\right)}^2$ ; comparing it with standard form of parabola equation $Y=-X^2$, opens downwards , has vertex (0,0) , axis of symmetry is X = 0.
• $y-8=Y;-2{\left(x-3\right)}^2=-{\left(\sqrt{2}\left(x-3\right)\right)}^2=-X^2$
• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y-8=Y=0;\sqrt{2}\left(x-3\right)=X=0\\ \Rightarrow y=8;x=3\\ \Rightarrow Vertex\equiv (3,8)\end{array}$

• Find Axis of Symmetry :

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

The equation of axis of symmetry is x = 3

• 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 $y-8=-2{\left(x-3\right)}^2$ , we will get the y -intercept.

  $\begin{array}{l}y-8=-2{\left(x-3\right)}^2\\ \Rightarrow y-8=-2{\left(0-3\right)}^2\mathrm{.............}[x=0]\\ \Rightarrow y-8=-2\left(9\right)\\ \Rightarrow y=-18+\mathrm{8.........................}[\text{add 8 each side]}\\ \Rightarrow y=-10\end{array}$

So, y -intercept = (0,-10)

• 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 $y-8=-2{\left(x-3\right)}^2$ , we will get the x -intercept.

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

So, x -intercepts are (1,0) and (5,0)

Use the above information to graph the equation $y-8=-2{\left(x-3\right)}^2$: