Find the Axis of Symmetry y=-2/3(x+5)^2-7

$y = - \frac{2}{3} {(x + 5)}^{2} - 7$

Step 1

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

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

$h = - 5$

$k = - 7$

Step 2

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

Opens Down

Step 3

Find the vertex $(h, k)$ .

$(- 5, - 7)$

Step 4

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.

$\frac{1}{4 \cdot (- \frac{2}{3})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{2}{3})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{2}{3})}$

Combine $4$ and $\frac{2}{3}$ .

$- \frac{1}{\frac{4 \cdot 2}{3}}$

Multiply $4$ by $2$ .

$- \frac{1}{\frac{8}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{8}))$

Multiply $\frac{3}{8}$ by $1$ .

$- \frac{3}{8}$

Step 5

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.

$(- 5, - \frac{59}{8})$

Step 6

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

$x = - 5$

Step 7