Graph f(x)=(4x+9)(4x-9)

$f (x) = (4 x + 9) (4 x - 9)$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $(4 x + 9) (4 x - 9)$ .

Simplify the expression.

Expand $(4 x + 9) (4 x - 9)$ using the FOIL Method.

Apply the distributive property.

$4 x (4 x - 9) + 9 (4 x - 9)$

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 9 + 9 (4 x - 9)$

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 9 + 9 (4 x) + 9 \cdot - 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 \cdot 4 x \cdot x + 4 x \cdot - 9 + 9 (4 x) + 9 \cdot - 9$

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

Move $x$ .

$4 \cdot 4 (x \cdot x) + 4 x \cdot - 9 + 9 (4 x) + 9 \cdot - 9$

Multiply $x$ by $x$ .

$4 \cdot 4 x^{2} + 4 x \cdot - 9 + 9 (4 x) + 9 \cdot - 9$

Multiply $4$ by $4$ .

$16 x^{2} + 4 x \cdot - 9 + 9 (4 x) + 9 \cdot - 9$

Multiply $- 9$ by $4$ .

$16 x^{2} - 36 x + 9 (4 x) + 9 \cdot - 9$

Multiply $4$ by $9$ .

$16 x^{2} - 36 x + 36 x + 9 \cdot - 9$

Multiply $9$ by $- 9$ .

$16 x^{2} - 36 x + 36 x - 81$

Add $- 36 x$ and $36 x$ .

$16 x^{2} + 0 - 81$

Add $16 x^{2}$ and $0$ .

$16 x^{2} - 81$

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

$a = 16$

$b = 0$

$c = - 81$

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

$d = \frac{0}{2 \cdot 16}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 (0)}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{0}{16}$

Cancel the common factor of $0$ and $16$ .

Factor $16$ out of $0$ .

$d = \frac{16 (0)}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot 0}{16 \cdot 1}$

Cancel the common factor.

$d = \frac{16 \cdot 0}{16 \cdot 1}$

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

$e = - 81 - \frac{0^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

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

$e = - 81 - \frac{0}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = - 81 - \frac{0}{64}$

Divide $0$ by $64$ .

$e = - 81 - 0$

Multiply $- 1$ by $0$ .

$e = - 81 + 0$

Add $- 81$ and $0$ .

$e = - 81$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x + 0)}^{2} - 81$ .

$16 {(x + 0)}^{2} - 81$

Set $y$ equal to the new right side.

$y = 16 {(x + 0)}^{2} - 81$

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

$a = 16$

$h = 0$

$k = - 81$

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

Opens Up

Find the vertex $(h, k)$ .

$(0, - 81)$

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.

$\frac{1}{4 \cdot 16}$

Multiply $4$ by $16$ .

$\frac{1}{64}$

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.

$(0, - \frac{5183}{64})$

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.

$y = - \frac{5185}{64}$

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

Direction: Opens Up

Vertex: $(0, - 81)$

Focus: $(0, - \frac{5183}{64})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{5185}{64}$

Direction: Opens Up

Vertex: $(0, - 81)$

Focus: $(0, - \frac{5183}{64})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{5185}{64}$

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.

$f (- 1) = (4 (- 1) + 9) (4 (- 1) - 9)$

Simplify the result.

Multiply $4$ by $- 1$ .

$f (- 1) = (- 4 + 9) (4 (- 1) - 9)$

Add $- 4$ and $9$ .

$f (- 1) = 5 (4 (- 1) - 9)$

Multiply $4$ by $- 1$ .

$f (- 1) = 5 (- 4 - 9)$

Subtract $9$ from $- 4$ .

$f (- 1) = 5 \cdot - 13$

Multiply $5$ by $- 13$ .

$f (- 1) = - 65$

The final answer is $- 65$ .

$- 65$

The $y$ value at $x = - 1$ is $- 65$ .

$y = - 65$

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

$f (1) = (4 (1) + 9) (4 (1) - 9)$

Simplify the result.

Multiply $4$ by $1$ .

$f (1) = (4 + 9) (4 (1) - 9)$

Add $4$ and $9$ .

$f (1) = 13 (4 (1) - 9)$

Multiply $4$ by $1$ .

$f (1) = 13 (4 - 9)$

Subtract $9$ from $4$ .

$f (1) = 13 \cdot - 5$

Multiply $13$ by $- 5$ .

$f (1) = - 65$

The final answer is $- 65$ .

$- 65$

The $y$ value at $x = 1$ is $- 65$ .

$y = - 65$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 65 \\ 0 & - 81 \\ 1 & - 65 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(0, - 81)$

Focus: $(0, - \frac{5183}{64})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{5185}{64}$

$\begin{array}{c} x & y \\ - 1 & - 65 \\ 0 & - 81 \\ 1 & - 65 \end{array}$

Step 4

Graph f(x)=(4x+9)(4x-9)

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