Graph f(x)=(x-4)(x+4)

$f (x) = (x - 4) (x + 4)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $(x - 4) (x + 4)$ .

Simplify the expression.

Expand $(x - 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$x (x + 4) - 4 (x + 4)$

Apply the distributive property.

$x \cdot x + x \cdot 4 - 4 (x + 4)$

Apply the distributive property.

$x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$

Combine the opposite terms in $x \cdot x + x \cdot 4 - 4 x - 4 \cdot 4$ .

Reorder the factors in the terms $x \cdot 4$ and $- 4 x$ .

$x \cdot x + 4 x - 4 x - 4 \cdot 4$

Subtract $4 x$ from $4 x$ .

$x \cdot x + 0 - 4 \cdot 4$

Add $x \cdot x$ and $0$ .

$x \cdot x - 4 \cdot 4$

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$x^{2} - 16$

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

$a = 1$

$b = 0$

$c = - 16$

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

$d = \frac{0}{2 \cdot 1}$

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 (0)}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{0}{1}$

Divide $0$ by $1$ .

$d = 0$

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

$e = - 16 - \frac{0^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = - 16 - \frac{0}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 16 - \frac{0}{4}$

Divide $0$ by $4$ .

$e = - 16 - 0$

Multiply $- 1$ by $0$ .

$e = - 16 + 0$

Add $- 16$ and $0$ .

$e = - 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 0)}^{2} - 16$ .

${(x + 0)}^{2} - 16$

Set $y$ equal to the new right side.

$y = {(x + 0)}^{2} - 16$

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

$a = 1$

$h = 0$

$k = - 16$

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

Opens Up

Find the vertex $(h, k)$ .

$(0, - 16)$

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.

$(0, - \frac{63}{4})$

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

$x = 0$

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.

$y = - \frac{65}{4}$

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

Direction: Opens Up

Vertex: $(0, - 16)$

Focus: $(0, - \frac{63}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{65}{4}$

Direction: Opens Up

Vertex: $(0, - 16)$

Focus: $(0, - \frac{63}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{65}{4}$

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 $- 1$ in the expression.

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

Simplify the result.

Subtract $4$ from $- 1$ .

$f (- 1) = - 5 ((- 1) + 4)$

Add $- 1$ and $4$ .

$f (- 1) = - 5 \cdot 3$

Multiply $- 5$ by $3$ .

$f (- 1) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = - 1$ is $- 15$ .

$y = - 15$

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

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

Simplify the result.

Subtract $4$ from $- 2$ .

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

Add $- 2$ and $4$ .

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

Multiply $- 6$ by $2$ .

$f (- 2) = - 12$

The final answer is $- 12$ .

$- 12$

The $y$ value at $x = - 2$ is $- 12$ .

$y = - 12$

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

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

Simplify the result.

Subtract $4$ from $1$ .

$f (1) = - 3 ((1) + 4)$

Add $1$ and $4$ .

$f (1) = - 3 \cdot 5$

Multiply $- 3$ by $5$ .

$f (1) = - 15$

The final answer is $- 15$ .

$- 15$

The $y$ value at $x = 1$ is $- 15$ .

$y = - 15$

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

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

Simplify the result.

Subtract $4$ from $2$ .

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

Add $2$ and $4$ .

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

Multiply $- 2$ by $6$ .

$f (2) = - 12$

The final answer is $- 12$ .

$- 12$

The $y$ value at $x = 2$ is $- 12$ .

$y = - 12$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - 12 \\ - 1 & - 15 \\ 0 & - 16 \\ 1 & - 15 \\ 2 & - 12 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(0, - 16)$

Focus: $(0, - \frac{63}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{65}{4}$

$\begin{array}{c} x & y \\ - 2 & - 12 \\ - 1 & - 15 \\ 0 & - 16 \\ 1 & - 15 \\ 2 & - 12 \end{array}$

Step 4

Graph f(x)=(x-4)(x+4)

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