Find the Axis of Symmetry f(x)=-1/3x^2-5x+3

$f (x) = - \frac{1}{3} x^{2} - 5 x + 3$

Step 1

Write $f (x) = - \frac{1}{3} x^{2} - 5 x + 3$ as an equation.

$y = - \frac{1}{3} x^{2} - 5 x + 3$

Step 2

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{3}$ .

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

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

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

$a = - \frac{1}{3}$

$b = - 5$

$c = 3$

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

$d = \frac{- 5}{2 (- \frac{1}{3})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{- 5}{- 2 (\frac{1}{3})}$

Combine $- 2$ and $\frac{1}{3}$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - 5 (- \frac{3}{2})$

Multiply $- 5 (- \frac{3}{2})$ .

Multiply $- 1$ by $- 5$ .

$d = 5 (\frac{3}{2})$

Combine $5$ and $\frac{3}{2}$ .

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

Multiply $5$ by $3$ .

$d = \frac{15}{2}$

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

$e = 3 - \frac{{(- 5)}^{2}}{4 (- \frac{1}{3})}$

Simplify the right side.

Simplify each term.

Raise $- 5$ to the power of $2$ .

$e = 3 - \frac{25}{4 (- \frac{1}{3})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 3 - \frac{25}{- 4 (\frac{1}{3})}$

Combine $- 4$ and $\frac{1}{3}$ .

$e = 3 - \frac{25}{\frac{- 4}{3}}$

Move the negative in front of the fraction.

$e = 3 - \frac{25}{- \frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 3 - (25 (- \frac{3}{4}))$

Multiply $25 (- \frac{3}{4})$ .

Multiply $- 1$ by $25$ .

$e = 3 - (- 25 (\frac{3}{4}))$

Combine $- 25$ and $\frac{3}{4}$ .

$e = 3 - \frac{- 25 \cdot 3}{4}$

Multiply $- 25$ by $3$ .

$e = 3 - \frac{- 75}{4}$

Move the negative in front of the fraction.

$e = 3 - - \frac{75}{4}$

Multiply $- - \frac{75}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = 3 + 1 (\frac{75}{4})$

Multiply $\frac{75}{4}$ by $1$ .

$e = 3 + \frac{75}{4}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 3 \cdot \frac{4}{4} + \frac{75}{4}$

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

$e = \frac{3 \cdot 4}{4} + \frac{75}{4}$

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 4 + 75}{4}$

Simplify the numerator.

Multiply $3$ by $4$ .

$e = \frac{12 + 75}{4}$

Add $12$ and $75$ .

$e = \frac{87}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{3} {(x + \frac{15}{2})}^{2} + \frac{87}{4}$ .

$- \frac{1}{3} {(x + \frac{15}{2})}^{2} + \frac{87}{4}$

Set $y$ equal to the new right side.

$y = - \frac{1}{3} \cdot {(x + \frac{15}{2})}^{2} + \frac{87}{4}$

Step 3

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

$a = - \frac{1}{3}$

$h = - \frac{15}{2}$

$k = \frac{87}{4}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- \frac{15}{2}, \frac{87}{4})$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

Combine $4$ and $\frac{1}{3}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{3}{4}$ by $1$ .

$- \frac{3}{4}$

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.

$(- \frac{15}{2}, 21)$

Step 8

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

$x = - \frac{15}{2}$

Step 9

Find the Axis of Symmetry f(x)=-1/3x^2-5x+3

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