Graph f(x)=(x-8)^2

$f (x) = {(x - 8)}^{2}$

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 $k$ .

$a = 1$

$h = 8$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(8, 0)$

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.

$(8, \frac{1}{4})$

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

$x = 8$

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: $(8, 0)$

Focus: $(8, \frac{1}{4})$

Axis of Symmetry: $x = 8$

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

Direction: Opens Up

Vertex: $(8, 0)$

Focus: $(8, \frac{1}{4})$

Axis of Symmetry: $x = 8$

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 $7$ in the expression.

$f (7) = {(7)}^{2} - 16 \cdot 7 + 64$

Simplify the result.

Simplify each term.

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

$f (7) = 49 - 16 \cdot 7 + 64$

Multiply $- 16$ by $7$ .

$f (7) = 49 - 112 + 64$

Simplify by adding and subtracting.

Subtract $112$ from $49$ .

$f (7) = - 63 + 64$

Add $- 63$ and $64$ .

$f (7) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 7$ is $1$ .

$y = 1$

Replace the variable $x$ with $6$ in the expression.

$f (6) = {(6)}^{2} - 16 \cdot 6 + 64$

Simplify the result.

Simplify each term.

Raise $6$ to the power of $2$ .

$f (6) = 36 - 16 \cdot 6 + 64$

Multiply $- 16$ by $6$ .

$f (6) = 36 - 96 + 64$

Simplify by adding and subtracting.

Subtract $96$ from $36$ .

$f (6) = - 60 + 64$

Add $- 60$ and $64$ .

$f (6) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 6$ is $4$ .

$y = 4$

Replace the variable $x$ with $9$ in the expression.

$f (9) = {(9)}^{2} - 16 \cdot 9 + 64$

Simplify the result.

Simplify each term.

Raise $9$ to the power of $2$ .

$f (9) = 81 - 16 \cdot 9 + 64$

Multiply $- 16$ by $9$ .

$f (9) = 81 - 144 + 64$

Simplify by adding and subtracting.

Subtract $144$ from $81$ .

$f (9) = - 63 + 64$

Add $- 63$ and $64$ .

$f (9) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = 9$ is $1$ .

$y = 1$

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

$f (10) = {(10)}^{2} - 16 \cdot 10 + 64$

Simplify the result.

Simplify each term.

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

$f (10) = 100 - 16 \cdot 10 + 64$

Multiply $- 16$ by $10$ .

$f (10) = 100 - 160 + 64$

Simplify by adding and subtracting.

Subtract $160$ from $100$ .

$f (10) = - 60 + 64$

Add $- 60$ and $64$ .

$f (10) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 10$ is $4$ .

$y = 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 6 & 4 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ 10 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(8, 0)$

Focus: $(8, \frac{1}{4})$

Axis of Symmetry: $x = 8$

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

$\begin{array}{c} x & y \\ 6 & 4 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ 10 & 4 \end{array}$

Step 4

Graph f(x)=(x-8)^2

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