Find the Axis of Symmetry y=3/-4x^2+6x+6

$y = \frac{3}{- 4} x^{2} + 6 x + 6$

Step 1

Rewrite the equation in vertex form.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{3}{4} x^{2} + 6 x + 6$

Combine $x^{2}$ and $\frac{3}{4}$ .

$y = - \frac{x^{2} \cdot 3}{4} + 6 x + 6$

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 x^{2}}{4} + 6 x + 6$

Complete the square for $- \frac{3 x^{2}}{4} + 6 x + 6$ .

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

$a = - \frac{3}{4}$

$b = 6$

$c = 6$

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

$d = \frac{6}{2 (- \frac{3}{4})}$

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 (- \frac{3}{4})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 (- \frac{3}{4})}$

Rewrite the expression.

$d = \frac{3}{- \frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 3 (- \frac{4}{3})$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$d = 3 (\frac{- 4}{3})$

Cancel the common factor.

$d = 3 (\frac{- 4}{3})$

Rewrite the expression.

$d = - 4$

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

$e = 6 - \frac{6^{2}}{4 (- \frac{3}{4})}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 6 - \frac{36}{4 (- \frac{3}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 6 - \frac{36}{- 4 (\frac{3}{4})}$

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

$e = 6 - \frac{36}{\frac{- 4 \cdot 3}{4}}$

Multiply $- 4$ by $3$ .

$e = 6 - \frac{36}{\frac{- 12}{4}}$

Divide $- 12$ by $4$ .

$e = 6 - \frac{36}{- 3}$

Divide $36$ by $- 3$ .

$e = 6 - - 12$

Multiply $- 1$ by $- 12$ .

$e = 6 + 12$

Add $6$ and $12$ .

$e = 18$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{4} {(x - 4)}^{2} + 18$ .

$- \frac{3}{4} {(x - 4)}^{2} + 18$

Set $y$ equal to the new right side.

$y = - \frac{3}{4} \cdot {(x - 4)}^{2} + 18$

Step 2

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

$a = - \frac{3}{4}$

$h = 4$

$k = 18$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(4, 18)$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

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

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

Simplify the expression.

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{4}}$

Divide $12$ by $4$ .

$- \frac{1}{3}$

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.

$(4, \frac{53}{3})$

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=3/-4x^2+6x+6

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