Find the Axis of Symmetry f(x)=(x+9)(x-21)

$f (x) = (x + 9) (x - 21)$

Step 1

Write $f (x) = (x + 9) (x - 21)$ as an equation.

$y = (x + 9) (x - 21)$

Step 2

Rewrite the equation in vertex form.

Complete the square for $(x + 9) (x - 21)$ .

Simplify the expression.

Expand $(x + 9) (x - 21)$ using the FOIL Method.

Apply the distributive property.

$x (x - 21) + 9 (x - 21)$

Apply the distributive property.

$x \cdot x + x \cdot - 21 + 9 (x - 21)$

Apply the distributive property.

$x \cdot x + x \cdot - 21 + 9 x + 9 \cdot - 21$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 21 + 9 x + 9 \cdot - 21$

Move $- 21$ to the left of $x$ .

$x^{2} - 21 \cdot x + 9 x + 9 \cdot - 21$

Multiply $9$ by $- 21$ .

$x^{2} - 21 x + 9 x - 189$

Add $- 21 x$ and $9 x$ .

$x^{2} - 12 x - 189$

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

$a = 1$

$b = - 12$

$c = - 189$

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

$d = \frac{- 12}{2 \cdot 1}$

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 6}{1}$

Divide $- 6$ by $1$ .

$d = - 6$

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

$e = - 189 - \frac{{(- 12)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = - 189 - \frac{144}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = - 189 - \frac{144}{4}$

Divide $144$ by $4$ .

$e = - 189 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = - 189 - 36$

Subtract $36$ from $- 189$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 6)}^{2} - 225$ .

${(x - 6)}^{2} - 225$

Set $y$ equal to the new right side.

$y = {(x - 6)}^{2} - 225$

Step 3

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

$a = 1$

$h = 6$

$k = - 225$

Step 4

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

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(6, - 225)$

Step 6

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.

$\frac{1}{4}$

Step 7

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.

$(6, - \frac{899}{4})$

Step 8

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

$x = 6$

Step 9

Find the Axis of Symmetry f(x)=(x+9)(x-21)

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