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

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

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

Simplify the expression.

Expand $(x - 3) (2 x + 2)$ using the FOIL Method.

Apply the distributive property.

$x (2 x + 2) - 3 (2 x + 2)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

$2 x^{2} + x \cdot 2 - 3 (2 x) - 3 \cdot 2$

Move $2$ to the left of $x$ .

$2 x^{2} + 2 \cdot x - 3 (2 x) - 3 \cdot 2$

Multiply $2$ by $- 3$ .

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

Multiply $- 3$ by $2$ .

$2 x^{2} + 2 x - 6 x - 6$

Subtract $6 x$ from $2 x$ .

$2 x^{2} - 4 x - 6$

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

$a = 2$

$b = - 4$

$c = - 6$

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

$d = \frac{- 4}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 2}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{- 2}{2}$

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = - 6 - \frac{{(- 4)}^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

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

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

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 2$ .

$e = - 6 - \frac{4 \cdot 4}{4 (2)}$

Cancel the common factor.

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

Rewrite the expression.

$e = - 6 - \frac{4}{2}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$e = - 6 - \frac{2 \cdot 2}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$e = - 6 - \frac{2 \cdot 2}{2 (1)}$

Cancel the common factor.

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

Rewrite the expression.

$e = - 6 - \frac{2}{1}$

Divide $2$ by $1$ .

$e = - 6 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = - 6 - 2$

Subtract $2$ from $- 6$ .

$e = - 8$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 1)}^{2} - 8$ .

$2 {(x - 1)}^{2} - 8$

Set $y$ equal to the new right side.

$y = 2 {(x - 1)}^{2} - 8$

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

$a = 2$

$h = 1$

$k = - 8$

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

Opens Up

Find the vertex $(h, k)$ .

$(1, - 8)$

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.

$\frac{1}{4 \cdot 2}$

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(1, - \frac{63}{8})$

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

$x = 1$

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.

$y = - \frac{65}{8}$

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

Direction: Opens Up

Vertex: $(1, - 8)$

Focus: $(1, - \frac{63}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{65}{8}$

Direction: Opens Up

Vertex: $(1, - 8)$

Focus: $(1, - \frac{63}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{65}{8}$

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 $0$ in the expression.

$f (0) = ((0) - 3) (2 (0) + 2)$

Simplify the result.

Subtract $3$ from $0$ .

$f (0) = - 3 (2 (0) + 2)$

Multiply $2$ by $0$ .

$f (0) = - 3 (0 + 2)$

Add $0$ and $2$ .

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

Multiply $- 3$ by $2$ .

$f (0) = - 6$

The final answer is $- 6$ .

$- 6$

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

$y = - 6$

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

$f (- 1) = ((- 1) - 3) (2 (- 1) + 2)$

Simplify the result.

Subtract $3$ from $- 1$ .

$f (- 1) = - 4 (2 (- 1) + 2)$

Multiply $2$ by $- 1$ .

$f (- 1) = - 4 (- 2 + 2)$

Add $- 2$ and $2$ .

$f (- 1) = - 4 \cdot 0$

Multiply $- 4$ by $0$ .

$f (- 1) = 0$

The final answer is $0$ .

$0$

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

$y = 0$

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

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

Simplify the result.

Subtract $3$ from $2$ .

$f (2) = - 1 (2 (2) + 2)$

Multiply $2$ by $2$ .

$f (2) = - 1 (4 + 2)$

Add $4$ and $2$ .

$f (2) = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$f (2) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = 2$ is $- 6$ .

$y = - 6$

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

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

Simplify the result.

Subtract $3$ from $3$ .

$f (3) = 0 (2 (3) + 2)$

Multiply $2$ by $3$ .

$f (3) = 0 (6 + 2)$

Add $6$ and $2$ .

$f (3) = 0 \cdot 8$

Multiply $0$ by $8$ .

$f (3) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = 3$ is $0$ .

$y = 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - 6 \\ 1 & - 8 \\ 2 & - 6 \\ 3 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, - 8)$

Focus: $(1, - \frac{63}{8})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{65}{8}$

$\begin{array}{c} x & y \\ - 1 & 0 \\ 0 & - 6 \\ 1 & - 8 \\ 2 & - 6 \\ 3 & 0 \end{array}$

Step 4

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

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