Find the Axis of Symmetry y=-5x^2+10x+12

$y = - 5 x^{2} + 10 x + 12$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 5 x^{2} + 10 x + 12$ .

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

$a = - 5$

$b = 10$

$c = 12$

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

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

Simplify the right side.

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $5$ out of $5$ .

$d = \frac{5 (1)}{- 5}$

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = 12 - \frac{10^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = 12 - \frac{100}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = 12 - \frac{100}{- 20}$

Divide $100$ by $- 20$ .

$e = 12 - - 5$

Multiply $- 1$ by $- 5$ .

$e = 12 + 5$

Add $12$ and $5$ .

$e = 17$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x - 1)}^{2} + 17$ .

$- 5 {(x - 1)}^{2} + 17$

Set $y$ equal to the new right side.

$y = - 5 {(x - 1)}^{2} + 17$

Step 2

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

$a = - 5$

$h = 1$

$k = 17$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(1, 17)$

Step 5

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 6

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.

$(1, \frac{339}{20})$

Step 7

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

$x = 1$

Step 8

Find the Axis of Symmetry y=-5x^2+10x+12

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