Chapter 7.5, Problem 12WE

To determine

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

Answer to Problem 12WE

Vertex : $(6,-3)$

Axis of Symmetry : x = 6

y -intercept : (0,10)

x -intercepts are (3,0) and (9,0)

Explanation of Solution

Given Information:

Given equation is :

  $y+3=\frac{1}{3}{\left(x-6\right)}^2$

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 $y+3=\frac{1}{3}{\left(x-6\right)}^2$ ; comparing it with standard form of parabola equation $Y=X^2$, opens upward , has vertex (0,0) , axis of symmetry is X = 0.

$y+3=Y;\frac{1}{3}{\left(x-6\right)}^2={\left(\frac{1}{\sqrt{3}}\left(x-6\right)\right)}^2=X^2$

Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y+3=Y=0;\frac{1}{\sqrt{3}}\left(x-6\right)=X=0\\ \Rightarrow y=-3;x=6\\ \Rightarrow Vertex\equiv (6,-3)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=X}}^2]\\ \Rightarrow \frac{1}{\sqrt{3}}\left(x-6\right)=\mathrm{0........}[X=\frac{1}{\sqrt{3}}\left(x-6\right)]\\ \Rightarrow x=\mathrm{6..................}[multiply\text{ each side by }\sqrt{3}\text{ then add 6 each side]}\end{array}$

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 $y+3=\frac{1}{3}{\left(x-6\right)}^2$ , we will get the y -intercept.

  $\begin{array}{l}y+3=\frac{1}{3}{\left(x-6\right)}^2\\ \Rightarrow y+3=\frac{1}{3}{\left(0-6\right)}^2\mathrm{.............}[x=0]\\ \Rightarrow y+3=\frac{1}{2}\left(36\right)\\ \Rightarrow y=13-\mathrm{3.........................}[subtract\text{ 3 from each side]}\\ \Rightarrow y=10\end{array}$

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 $y+3=\frac{1}{3}{\left(x-6\right)}^2$ , we will get the x -intercept.

  $\begin{array}{l}y+3=\frac{1}{3}{\left(x-6\right)}^2\\ \Rightarrow 0+3=\frac{1}{3}{\left(x-6\right)}^2\mathrm{.............}[y=0]\\ \Rightarrow 3=\frac{1}{3}{\left(x-6\right)}^2\\ \Rightarrow {\left(x-6\right)}^2=\mathrm{9....................}[\text{multiply each side by 3]}\\ \Rightarrow x-6=\pm \mathrm{3.......................}[\text{take square root each side]}\\ \Rightarrow x=6\pm \mathrm{3......................}[add\text{ 6 each side]}\\ \Rightarrow x=9\text{ and }x=3\end{array}$

So, x -intercepts are (3,0) and (9,0).

Use the above information to graph the equation $y+3=\frac{1}{3}{\left(x-6\right)}^2$: