Graph y=(x-9)^2

$y = {(x - 9)}^{2}$

Step 1

Find the properties of the given parabola.

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = 9$

$k = 0$

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Find the vertex $(h, k)$ .

$(9, 0)$

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$\frac{1}{4 \cdot 1}$

Rewrite the expression.

$\frac{1}{4}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(9, \frac{1}{4})$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 9$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{1}{4}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(9, 0)$

Focus: $(9, \frac{1}{4})$

Axis of Symmetry: $x = 9$

Directrix: $y = - \frac{1}{4}$

Direction: Opens Up

Vertex: $(9, 0)$

Focus: $(9, \frac{1}{4})$

Axis of Symmetry: $x = 9$

Directrix: $y = - \frac{1}{4}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} - 18 \cdot 8 + 81$

Simplify the result.

Simplify each term.

Raise $8$ to the power of $2$ .

$f (8) = 64 - 18 \cdot 8 + 81$

Multiply $- 18$ by $8$ .

$f (8) = 64 - 144 + 81$

Simplify by adding and subtracting.

Subtract $144$ from $64$ .

$f (8) = - 80 + 81$

Add $- 80$ and $81$ .

$f (8) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 8$ is $1$ .

$y = 1$

Replace the variable $x$ with $7$ in the expression.

$f (7) = {(7)}^{2} - 18 \cdot 7 + 81$

Simplify the result.

Simplify each term.

Raise $7$ to the power of $2$ .

$f (7) = 49 - 18 \cdot 7 + 81$

Multiply $- 18$ by $7$ .

$f (7) = 49 - 126 + 81$

Simplify by adding and subtracting.

Subtract $126$ from $49$ .

$f (7) = - 77 + 81$

Add $- 77$ and $81$ .

$f (7) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 7$ is $4$ .

$y = 4$

Replace the variable $x$ with $10$ in the expression.

$f (10) = {(10)}^{2} - 18 \cdot 10 + 81$

Simplify the result.

Simplify each term.

Raise $10$ to the power of $2$ .

$f (10) = 100 - 18 \cdot 10 + 81$

Multiply $- 18$ by $10$ .

$f (10) = 100 - 180 + 81$

Simplify by adding and subtracting.

Subtract $180$ from $100$ .

$f (10) = - 80 + 81$

Add $- 80$ and $81$ .

$f (10) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 10$ is $1$ .

$y = 1$

Replace the variable $x$ with $11$ in the expression.

$f (11) = {(11)}^{2} - 18 \cdot 11 + 81$

Simplify the result.

Simplify each term.

Raise $11$ to the power of $2$ .

$f (11) = 121 - 18 \cdot 11 + 81$

Multiply $- 18$ by $11$ .

$f (11) = 121 - 198 + 81$

Simplify by adding and subtracting.

Subtract $198$ from $121$ .

$f (11) = - 77 + 81$

Add $- 77$ and $81$ .

$f (11) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 11$ is $4$ .

$y = 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 7 & 4 \\ 8 & 1 \\ 9 & 0 \\ 10 & 1 \\ 11 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(9, 0)$

Focus: $(9, \frac{1}{4})$

Axis of Symmetry: $x = 9$

Directrix: $y = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 7 & 4 \\ 8 & 1 \\ 9 & 0 \\ 10 & 1 \\ 11 & 4 \end{array}$

Step 4

Graph y=(x-9)^2

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