Find the Axis of Symmetry f(x)=-5x^2+3x+2

$f (x) = - 5 x^{2} + 3 x + 2$

Step 1

Write $f (x) = - 5 x^{2} + 3 x + 2$ as an equation.

$y = - 5 x^{2} + 3 x + 2$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 5 x^{2} + 3 x + 2$ .

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

$a = - 5$

$b = 3$

$c = 2$

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

$d = \frac{3}{2 \cdot - 5}$

Simplify the right side.

Multiply $2$ by $- 5$ .

$d = \frac{3}{- 10}$

Move the negative in front of the fraction.

$d = - \frac{3}{10}$

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

$e = 2 - \frac{3^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = 2 - \frac{9}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = 2 - \frac{9}{- 20}$

Move the negative in front of the fraction.

$e = 2 - - \frac{9}{20}$

Multiply $- - \frac{9}{20}$ .

Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{9}{20})$

Multiply $\frac{9}{20}$ by $1$ .

$e = 2 + \frac{9}{20}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$e = 2 \cdot \frac{20}{20} + \frac{9}{20}$

Combine $2$ and $\frac{20}{20}$ .

$e = \frac{2 \cdot 20}{20} + \frac{9}{20}$

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 20 + 9}{20}$

Simplify the numerator.

Multiply $2$ by $20$ .

$e = \frac{40 + 9}{20}$

Add $40$ and $9$ .

$e = \frac{49}{20}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x - \frac{3}{10})}^{2} + \frac{49}{20}$ .

$- 5 {(x - \frac{3}{10})}^{2} + \frac{49}{20}$

Set $y$ equal to the new right side.

$y = - 5 {(x - \frac{3}{10})}^{2} + \frac{49}{20}$

Step 3

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

$a = - 5$

$h = \frac{3}{10}$

$k = \frac{49}{20}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(\frac{3}{10}, \frac{49}{20})$

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.

$(\frac{3}{10}, \frac{12}{5})$

Step 8

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

$x = \frac{3}{10}$

Step 9

Find the Axis of Symmetry f(x)=-5x^2+3x+2

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