Graph f(x)=x^2+14x+7

$f (x) = x^{2} + 14 x + 7$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $x^{2} + 14 x + 7$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 14$

$c = 7$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{14}{2 \cdot 1}$

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 7}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{7}{1}$

Divide $7$ by $1$ .

$d = 7$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 7 - \frac{14^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $14$ to the power of $2$ .

$e = 7 - \frac{196}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 7 - \frac{196}{4}$

Divide $196$ by $4$ .

$e = 7 - 1 \cdot 49$

Multiply $- 1$ by $49$ .

$e = 7 - 49$

Subtract $49$ from $7$ .

$e = - 42$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 7)}^{2} - 42$ .

${(x + 7)}^{2} - 42$

Set $y$ equal to the new right side.

$y = {(x + 7)}^{2} - 42$

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

$a = 1$

$h = - 7$

$k = - 42$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 7, - 42)$

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.

$(- 7, - \frac{167}{4})$

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.

$y = - \frac{169}{4}$

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

Direction: Opens Up

Vertex: $(- 7, - 42)$

Focus: $(- 7, - \frac{167}{4})$

Axis of Symmetry: $x = - 7$

Directrix: $y = - \frac{169}{4}$

Direction: Opens Up

Vertex: $(- 7, - 42)$

Focus: $(- 7, - \frac{167}{4})$

Axis of Symmetry: $x = - 7$

Directrix: $y = - \frac{169}{4}$

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.

$f (- 8) = {(- 8)}^{2} + 14 (- 8) + 7$

Simplify the result.

Simplify each term.

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

$f (- 8) = 64 + 14 (- 8) + 7$

Multiply $14$ by $- 8$ .

$f (- 8) = 64 - 112 + 7$

Simplify by adding and subtracting.

Subtract $112$ from $64$ .

$f (- 8) = - 48 + 7$

Add $- 48$ and $7$ .

$f (- 8) = - 41$

The final answer is $- 41$ .

$- 41$

The $y$ value at $x = - 8$ is $- 41$ .

$y = - 41$

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

$f (- 9) = {(- 9)}^{2} + 14 (- 9) + 7$

Simplify the result.

Simplify each term.

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

$f (- 9) = 81 + 14 (- 9) + 7$

Multiply $14$ by $- 9$ .

$f (- 9) = 81 - 126 + 7$

Simplify by adding and subtracting.

Subtract $126$ from $81$ .

$f (- 9) = - 45 + 7$

Add $- 45$ and $7$ .

$f (- 9) = - 38$

The final answer is $- 38$ .

$- 38$

The $y$ value at $x = - 9$ is $- 38$ .

$y = - 38$

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

$f (- 6) = {(- 6)}^{2} + 14 (- 6) + 7$

Simplify the result.

Simplify each term.

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

$f (- 6) = 36 + 14 (- 6) + 7$

Multiply $14$ by $- 6$ .

$f (- 6) = 36 - 84 + 7$

Simplify by adding and subtracting.

Subtract $84$ from $36$ .

$f (- 6) = - 48 + 7$

Add $- 48$ and $7$ .

$f (- 6) = - 41$

The final answer is $- 41$ .

$- 41$

The $y$ value at $x = - 6$ is $- 41$ .

$y = - 41$

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

$f (- 5) = {(- 5)}^{2} + 14 (- 5) + 7$

Simplify the result.

Simplify each term.

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

$f (- 5) = 25 + 14 (- 5) + 7$

Multiply $14$ by $- 5$ .

$f (- 5) = 25 - 70 + 7$

Simplify by adding and subtracting.

Subtract $70$ from $25$ .

$f (- 5) = - 45 + 7$

Add $- 45$ and $7$ .

$f (- 5) = - 38$

The final answer is $- 38$ .

$- 38$

The $y$ value at $x = - 5$ is $- 38$ .

$y = - 38$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 9 & - 38 \\ - 8 & - 41 \\ - 7 & - 42 \\ - 6 & - 41 \\ - 5 & - 38 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 7, - 42)$

Focus: $(- 7, - \frac{167}{4})$

Axis of Symmetry: $x = - 7$

Directrix: $y = - \frac{169}{4}$

$\begin{array}{c} x & y \\ - 9 & - 38 \\ - 8 & - 41 \\ - 7 & - 42 \\ - 6 & - 41 \\ - 5 & - 38 \end{array}$

Step 4

Graph f(x)=x^2+14x+7

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