Algebra Examples

$f (x) = x^{2} + 6 x + 5$

Step 1

Find the properties of the given parabola.

Tap for more steps...

Step 1.1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1.1

Complete the square for $x^{2} + 6 x + 5$ .

Tap for more steps...

Step 1.1.1.1

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

$a = 1$

$b = 6$

$c = 5$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

Tap for more steps...

Step 1.1.1.3.1

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

$d = \frac{6}{2 \cdot 1}$

Step 1.1.1.3.2

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 1.1.1.3.2.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot 1}$

Step 1.1.1.3.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.1.3.2.2.1

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

$d = \frac{2 \cdot 3}{2 (1)}$

Step 1.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot 1}$

Step 1.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{3}{1}$

Step 1.1.1.3.2.2.4

Divide $3$ by $1$ .

$d = 3$

Step 1.1.1.4

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

Tap for more steps...

Step 1.1.1.4.1

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

$e = 5 - \frac{6^{2}}{4 \cdot 1}$

Step 1.1.1.4.2

Simplify the right side.

Tap for more steps...

Step 1.1.1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.1.1.4.2.1.1

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

$e = 5 - \frac{36}{4 \cdot 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 5 - \frac{36}{4}$

Step 1.1.1.4.2.1.3

Divide $36$ by $4$ .

$e = 5 - 1 \cdot 9$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 5 - 9$

Step 1.1.1.4.2.2

Subtract $9$ from $5$ .

$e = - 4$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 3)}^{2} - 4$ .

${(x + 3)}^{2} - 4$

Step 1.1.2

Set $y$ equal to the new right side.

$y = {(x + 3)}^{2} - 4$

Step 1.2

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

$a = 1$

$h = - 3$

$k = - 4$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 3, - 4)$

Step 1.5

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

Tap for more steps...

Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Cancel the common factor of $1$ .

Tap for more steps...

Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.6

Find the focus.

Tap for more steps...

Step 1.6.1

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

$(h, k + p)$

Step 1.6.2

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

$(- 3, - \frac{15}{4})$

Step 1.7

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

$x = - 3$

Step 1.8

Find the directrix.

Tap for more steps...

Step 1.8.1

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.

$y = k - p$

Step 1.8.2

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

$y = - \frac{17}{4}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 3, - 4)$

Focus: $(- 3, - \frac{15}{4})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{17}{4}$

Direction: Opens Up

Vertex: $(- 3, - 4)$

Focus: $(- 3, - \frac{15}{4})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{17}{4}$

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.

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = {(- 4)}^{2} + 6 (- 4) + 5$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = 16 + 6 (- 4) + 5$

Step 2.2.1.2

Multiply $6$ by $- 4$ .

$f (- 4) = 16 - 24 + 5$

Step 2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.2.1

Subtract $24$ from $16$ .

$f (- 4) = - 8 + 5$

Step 2.2.2.2

Add $- 8$ and $5$ .

$f (- 4) = - 3$

Step 2.2.3

The final answer is $- 3$ .

$- 3$

Step 2.3

The $y$ value at $x = - 4$ is $- 3$ .

$y = - 3$

Step 2.4

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

$f (- 5) = {(- 5)}^{2} + 6 (- 5) + 5$

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

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

$f (- 5) = 25 + 6 (- 5) + 5$

Step 2.5.1.2

Multiply $6$ by $- 5$ .

$f (- 5) = 25 - 30 + 5$

Step 2.5.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.5.2.1

Subtract $30$ from $25$ .

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

Step 2.5.2.2

Add $- 5$ and $5$ .

$f (- 5) = 0$

Step 2.5.3

The final answer is $0$ .

$0$

Step 2.6

The $y$ value at $x = - 5$ is $0$ .

$y = 0$

Step 2.7

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = {(- 2)}^{2} + 6 (- 2) + 5$

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Simplify each term.

Tap for more steps...

Step 2.8.1.1

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

$f (- 2) = 4 + 6 (- 2) + 5$

Step 2.8.1.2

Multiply $6$ by $- 2$ .

$f (- 2) = 4 - 12 + 5$

Step 2.8.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.8.2.1

Subtract $12$ from $4$ .

$f (- 2) = - 8 + 5$

Step 2.8.2.2

Add $- 8$ and $5$ .

$f (- 2) = - 3$

Step 2.8.3

The final answer is $- 3$ .

$- 3$

Step 2.9

The $y$ value at $x = - 2$ is $- 3$ .

$y = - 3$

Step 2.10

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = {(- 1)}^{2} + 6 (- 1) + 5$

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Simplify each term.

Tap for more steps...

Step 2.11.1.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = 1 + 6 (- 1) + 5$

Step 2.11.1.2

Multiply $6$ by $- 1$ .

$f (- 1) = 1 - 6 + 5$

Step 2.11.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.11.2.1

Subtract $6$ from $1$ .

$f (- 1) = - 5 + 5$

Step 2.11.2.2

Add $- 5$ and $5$ .

$f (- 1) = 0$

Step 2.11.3

The final answer is $0$ .

$0$

Step 2.12

The $y$ value at $x = - 1$ is $0$ .

$y = 0$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & 0 \\ - 4 & - 3 \\ - 3 & - 4 \\ - 2 & - 3 \\ - 1 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, - 4)$

Focus: $(- 3, - \frac{15}{4})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{17}{4}$

$\begin{array}{c} x & y \\ - 5 & 0 \\ - 4 & - 3 \\ - 3 & - 4 \\ - 2 & - 3 \\ - 1 & 0 \end{array}$

Step 4