Find the Axis of Symmetry y=5x^2-40x-27

$y = 5 x^{2} - 40 x - 27$

Step 1

Rewrite the equation in vertex form.

Complete the square for $5 x^{2} - 40 x - 27$ .

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

$a = 5$

$b = - 40$

$c = - 27$

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

$d = \frac{- 40}{2 \cdot 5}$

Simplify the right side.

Cancel the common factor of $- 40$ and $2$ .

Factor $2$ out of $- 40$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 20$ and $5$ .

Factor $5$ out of $- 20$ .

$d = \frac{5 \cdot - 4}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$d = \frac{5 \cdot - 4}{5 (1)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = - 27 - \frac{{(- 40)}^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

Raise $- 40$ to the power of $2$ .

$e = - 27 - \frac{1600}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = - 27 - \frac{1600}{20}$

Divide $1600$ by $20$ .

$e = - 27 - 1 \cdot 80$

Multiply $- 1$ by $80$ .

$e = - 27 - 80$

Subtract $80$ from $- 27$ .

$e = - 107$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x - 4)}^{2} - 107$ .

$5 {(x - 4)}^{2} - 107$

Set $y$ equal to the new right side.

$y = 5 {(x - 4)}^{2} - 107$

Step 2

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

$a = 5$

$h = 4$

$k = - 107$

Step 3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(4, - 107)$

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.

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

Multiply $4$ by $5$ .

$\frac{1}{20}$

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.

$(4, - \frac{2139}{20})$

Step 7

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

$x = 4$

Step 8

Find the Axis of Symmetry y=5x^2-40x-27

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