Graph y=3(x+2)(x+6)

$y = 3 (x + 2) (x + 6)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $3 (x + 2) (x + 6)$ .

Simplify the expression.

Apply the distributive property.

$(3 x + 3 \cdot 2) (x + 6)$

Multiply $3$ by $2$ .

$(3 x + 6) (x + 6)$

Expand $(3 x + 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$3 x (x + 6) + 6 (x + 6)$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 6 + 6 (x + 6)$

Apply the distributive property.

$3 x \cdot x + 3 x \cdot 6 + 6 x + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 (x \cdot x) + 3 x \cdot 6 + 6 x + 6 \cdot 6$

Multiply $x$ by $x$ .

$3 x^{2} + 3 x \cdot 6 + 6 x + 6 \cdot 6$

Multiply $6$ by $3$ .

$3 x^{2} + 18 x + 6 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$3 x^{2} + 18 x + 6 x + 36$

Add $18 x$ and $6 x$ .

$3 x^{2} + 24 x + 36$

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

$a = 3$

$b = 24$

$c = 36$

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 $d = \frac{b}{2 a}$ .

$d = \frac{24}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot 12}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{12}{3}$

Cancel the common factor of $12$ and $3$ .

Factor $3$ out of $12$ .

$d = \frac{3 \cdot 4}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$d = \frac{3 \cdot 4}{3 (1)}$

Cancel the common factor.

$d = \frac{3 \cdot 4}{3 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 36 - \frac{24^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = 36 - \frac{576}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 36 - \frac{576}{12}$

Divide $576$ by $12$ .

$e = 36 - 1 \cdot 48$

Multiply $- 1$ by $48$ .

$e = 36 - 48$

Subtract $48$ from $36$ .

$e = - 12$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 4)}^{2} - 12$ .

$3 {(x + 4)}^{2} - 12$

Set $y$ equal to the new right side.

$y = 3 {(x + 4)}^{2} - 12$

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

$a = 3$

$h = - 4$

$k = - 12$

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

Opens Up

Find the vertex $(h, k)$ .

$(- 4, - 12)$

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.

$\frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$\frac{1}{12}$

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.

$(- 4, - \frac{143}{12})$

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

$x = - 4$

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.

$y = - \frac{145}{12}$

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

Direction: Opens Up

Vertex: $(- 4, - 12)$

Focus: $(- 4, - \frac{143}{12})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{145}{12}$

Direction: Opens Up

Vertex: $(- 4, - 12)$

Focus: $(- 4, - \frac{143}{12})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{145}{12}$

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 $- 5$ in the expression.

$f (- 5) = 3 ((- 5) + 2) ((- 5) + 6)$

Simplify the result.

Add $- 5$ and $2$ .

$f (- 5) = 3 \cdot (- 3 ((- 5) + 6))$

Multiply $3$ by $- 3$ .

$f (- 5) = - 9 ((- 5) + 6)$

Add $- 5$ and $6$ .

$f (- 5) = - 9 \cdot 1$

Multiply $- 9$ by $1$ .

$f (- 5) = - 9$

The final answer is $- 9$ .

$- 9$

The $y$ value at $x = - 5$ is $- 9$ .

$y = - 9$

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

$f (- 6) = 3 ((- 6) + 2) ((- 6) + 6)$

Simplify the result.

Add $- 6$ and $2$ .

$f (- 6) = 3 \cdot (- 4 ((- 6) + 6))$

Multiply $3$ by $- 4$ .

$f (- 6) = - 12 ((- 6) + 6)$

Add $- 6$ and $6$ .

$f (- 6) = - 12 \cdot 0$

Multiply $- 12$ by $0$ .

$f (- 6) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 6$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 3 ((- 3) + 2) ((- 3) + 6)$

Simplify the result.

Add $- 3$ and $2$ .

$f (- 3) = 3 \cdot (- 1 ((- 3) + 6))$

Multiply $3$ by $- 1$ .

$f (- 3) = - 3 ((- 3) + 6)$

Add $- 3$ and $6$ .

$f (- 3) = - 3 \cdot 3$

Multiply $- 3$ by $3$ .

$f (- 3) = - 9$

The final answer is $- 9$ .

$- 9$

The $y$ value at $x = - 3$ is $- 9$ .

$y = - 9$

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

$f (- 2) = 3 ((- 2) + 2) ((- 2) + 6)$

Simplify the result.

Add $- 2$ and $2$ .

$f (- 2) = 3 \cdot (0 ((- 2) + 6))$

Multiply $3$ by $0$ .

$f (- 2) = 0 ((- 2) + 6)$

Add $- 2$ and $6$ .

$f (- 2) = 0 \cdot 4$

Multiply $0$ by $4$ .

$f (- 2) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 2$ is $0$ .

$y = 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & 0 \\ - 5 & - 9 \\ - 4 & - 12 \\ - 3 & - 9 \\ - 2 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 4, - 12)$

Focus: $(- 4, - \frac{143}{12})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{145}{12}$

$\begin{array}{c} x & y \\ - 6 & 0 \\ - 5 & - 9 \\ - 4 & - 12 \\ - 3 & - 9 \\ - 2 & 0 \end{array}$

Step 4

Graph y=3(x+2)(x+6)

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