Graph h(x)=(x+7)^2

$h (x) = {(x + 7)}^{2}$

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 $k$ .

$a = 1$

$h = - 7$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 7, 0)$

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.

$(- 7, \frac{1}{4})$

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

$x = - 7$

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: $(- 7, 0)$

Focus: $(- 7, \frac{1}{4})$

Axis of Symmetry: $x = - 7$

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

Direction: Opens Up

Vertex: $(- 7, 0)$

Focus: $(- 7, \frac{1}{4})$

Axis of Symmetry: $x = - 7$

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.

$f (- 8) = {(- 8)}^{2} + 14 (- 8) + 49$

Simplify the result.

Simplify each term.

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

$f (- 8) = 64 + 14 (- 8) + 49$

Multiply $14$ by $- 8$ .

$f (- 8) = 64 - 112 + 49$

Simplify by adding and subtracting.

Subtract $112$ from $64$ .

$f (- 8) = - 48 + 49$

Add $- 48$ and $49$ .

$f (- 8) = 1$

The final answer is $1$ .

$1$

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

$y = 1$

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

$f (- 9) = {(- 9)}^{2} + 14 (- 9) + 49$

Simplify the result.

Simplify each term.

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

$f (- 9) = 81 + 14 (- 9) + 49$

Multiply $14$ by $- 9$ .

$f (- 9) = 81 - 126 + 49$

Simplify by adding and subtracting.

Subtract $126$ from $81$ .

$f (- 9) = - 45 + 49$

Add $- 45$ and $49$ .

$f (- 9) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = - 9$ is $4$ .

$y = 4$

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

$f (- 6) = {(- 6)}^{2} + 14 (- 6) + 49$

Simplify the result.

Simplify each term.

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

$f (- 6) = 36 + 14 (- 6) + 49$

Multiply $14$ by $- 6$ .

$f (- 6) = 36 - 84 + 49$

Simplify by adding and subtracting.

Subtract $84$ from $36$ .

$f (- 6) = - 48 + 49$

Add $- 48$ and $49$ .

$f (- 6) = 1$

The final answer is $1$ .

$1$

The $y$ value at $x = - 6$ is $1$ .

$y = 1$

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = {(- 5)}^{2} + 14 (- 5) + 49$

Simplify the result.

Simplify each term.

Raise $- 5$ to the power of $2$ .

$f (- 5) = 25 + 14 (- 5) + 49$

Multiply $14$ by $- 5$ .

$f (- 5) = 25 - 70 + 49$

Simplify by adding and subtracting.

Subtract $70$ from $25$ .

$f (- 5) = - 45 + 49$

Add $- 45$ and $49$ .

$f (- 5) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = - 5$ is $4$ .

$y = 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 9 & 4 \\ - 8 & 1 \\ - 7 & 0 \\ - 6 & 1 \\ - 5 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 7, 0)$

Focus: $(- 7, \frac{1}{4})$

Axis of Symmetry: $x = - 7$

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

$\begin{array}{c} x & y \\ - 9 & 4 \\ - 8 & 1 \\ - 7 & 0 \\ - 6 & 1 \\ - 5 & 4 \end{array}$

Step 4

Graph h(x)=(x+7)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.