Find the Axis of Symmetry f(x)=-x^2-3x+18

$f (x) = - x^{2} - 3 x + 18$

Step 1

Write $f (x) = - x^{2} - 3 x + 18$ as an equation.

$y = - x^{2} - 3 x + 18$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- x^{2} - 3 x + 18$ .

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

$a = - 1$

$b = - 3$

$c = 18$

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

$d = \frac{- 3}{2 \cdot - 1}$

Simplify the right side.

Multiply $2$ by $- 1$ .

$d = \frac{- 3}{- 2}$

Dividing two negative values results in a positive value.

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

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

$e = 18 - \frac{{(- 3)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$e = 18 - \frac{9}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 18 - \frac{9}{- 4}$

Move the negative in front of the fraction.

$e = 18 - - \frac{9}{4}$

Multiply $- - \frac{9}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 18 + 1 (\frac{9}{4})$

Multiply $\frac{9}{4}$ by $1$ .

$e = 18 + \frac{9}{4}$

To write $18$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 18 \cdot \frac{4}{4} + \frac{9}{4}$

Combine $18$ and $\frac{4}{4}$ .

$e = \frac{18 \cdot 4}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{18 \cdot 4 + 9}{4}$

Simplify the numerator.

Multiply $18$ by $4$ .

$e = \frac{72 + 9}{4}$

Add $72$ and $9$ .

$e = \frac{81}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + \frac{3}{2})}^{2} + \frac{81}{4}$ .

$- {(x + \frac{3}{2})}^{2} + \frac{81}{4}$

Set $y$ equal to the new right side.

$y = - {(x + \frac{3}{2})}^{2} + \frac{81}{4}$

Step 3

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

$a = - 1$

$h = - \frac{3}{2}$

$k = \frac{81}{4}$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, \frac{81}{4})$

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 $1$ and $- 1$ .

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

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

$(- \frac{3}{2}, 20)$

Step 8

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

$x = - \frac{3}{2}$

Step 9

Find the Axis of Symmetry f(x)=-x^2-3x+18

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