Find the Axis of Symmetry f(x)=-5x^2-20x-10

$f (x) = - 5 x^{2} - 20 x - 10$

Step 1

Write $f (x) = - 5 x^{2} - 20 x - 10$ as an equation.

$y = - 5 x^{2} - 20 x - 10$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 5 x^{2} - 20 x - 10$ .

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

$a = - 5$

$b = - 20$

$c = - 10$

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

$d = \frac{- 20}{2 \cdot - 5}$

Simplify the right side.

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20$ .

$d = \frac{2 \cdot - 10}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

$d = \frac{2 \cdot - 10}{2 (- 5)}$

Cancel the common factor.

$d = \frac{2 \cdot - 10}{2 \cdot - 5}$

Rewrite the expression.

$d = \frac{- 10}{- 5}$

Cancel the common factor of $- 10$ and $- 5$ .

Factor $- 5$ out of $- 10$ .

$d = \frac{- 5 (2)}{- 5}$

Cancel the common factors.

Factor $- 5$ out of $- 5$ .

$d = \frac{- 5 \cdot 2}{- 5 \cdot 1}$

Cancel the common factor.

$d = \frac{- 5 \cdot 2}{- 5 \cdot 1}$

Rewrite the expression.

$d = \frac{2}{1}$

Divide $2$ by $1$ .

$d = 2$

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

$e = - 10 - \frac{{(- 20)}^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $- 20$ to the power of $2$ .

$e = - 10 - \frac{400}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = - 10 - \frac{400}{- 20}$

Divide $400$ by $- 20$ .

$e = - 10 - - 20$

Multiply $- 1$ by $- 20$ .

$e = - 10 + 20$

Add $- 10$ and $20$ .

$e = 10$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x + 2)}^{2} + 10$ .

$- 5 {(x + 2)}^{2} + 10$

Set $y$ equal to the new right side.

$y = - 5 {(x + 2)}^{2} + 10$

Step 3

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

$a = - 5$

$h = - 2$

$k = 10$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 2, 10)$

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.

$\frac{1}{4 \cdot - 5}$

Simplify.

Multiply $4$ by $- 5$ .

$\frac{1}{- 20}$

Move the negative in front of the fraction.

$- \frac{1}{20}$

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.

$(- 2, \frac{199}{20})$

Step 8

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

$x = - 2$

Step 9

Find the Axis of Symmetry f(x)=-5x^2-20x-10

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