Find the Axis of Symmetry q(x)=-2x^2+6x+9

$q (x) = - 2 x^{2} + 6 x + 9$

Step 1

Write $q (x) = - 2 x^{2} + 6 x + 9$ as an equation.

$y = - 2 x^{2} + 6 x + 9$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} + 6 x + 9$ .

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

$a = - 2$

$b = 6$

$c = 9$

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

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

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

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

$d = \frac{2 \cdot 3}{2 (- 2)}$

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$e = 9 - \frac{6^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 9 - \frac{36}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = 9 - \frac{36}{- 8}$

Cancel the common factor of $36$ and $- 8$ .

Factor $4$ out of $36$ .

$e = 9 - \frac{4 (9)}{- 8}$

Cancel the common factors.

Factor $4$ out of $- 8$ .

$e = 9 - \frac{4 \cdot 9}{4 \cdot - 2}$

Cancel the common factor.

$e = 9 - \frac{4 \cdot 9}{4 \cdot - 2}$

Rewrite the expression.

$e = 9 - \frac{9}{- 2}$

Move the negative in front of the fraction.

$e = 9 - - \frac{9}{2}$

Multiply $- - \frac{9}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = 9 + 1 (\frac{9}{2})$

Multiply $\frac{9}{2}$ by $1$ .

$e = 9 + \frac{9}{2}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 9 \cdot \frac{2}{2} + \frac{9}{2}$

Combine $9$ and $\frac{2}{2}$ .

$e = \frac{9 \cdot 2}{2} + \frac{9}{2}$

Combine the numerators over the common denominator.

$e = \frac{9 \cdot 2 + 9}{2}$

Simplify the numerator.

Multiply $9$ by $2$ .

$e = \frac{18 + 9}{2}$

Add $18$ and $9$ .

$e = \frac{27}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$ .

$- 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$

Set $y$ equal to the new right side.

$y = - 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 3

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

$a = - 2$

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

$k = \frac{27}{2}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{27}{2})$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(\frac{3}{2}, \frac{107}{8})$

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 q(x)=-2x^2+6x+9

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