Find the Axis of Symmetry y=9x^2-12x+4

$y = 9 x^{2} - 12 x + 4$

Step 1

Rewrite the equation in vertex form.

Complete the square for $9 x^{2} - 12 x + 4$ .

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

$a = 9$

$b = - 12$

$c = 4$

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

$d = \frac{- 12}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 6}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 6}{9}$

Cancel the common factor of $- 6$ and $9$ .

Factor $3$ out of $- 6$ .

$d = \frac{3 (- 2)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$e = 4 - \frac{{(- 12)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = 4 - \frac{144}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 4 - \frac{144}{36}$

Divide $144$ by $36$ .

$e = 4 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 4 - 4$

Subtract $4$ from $4$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - \frac{2}{3})}^{2} + 0$ .

$9 {(x - \frac{2}{3})}^{2} + 0$

Set $y$ equal to the new right side.

$y = 9 {(x - \frac{2}{3})}^{2} + 0$

Step 2

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

$a = 9$

$h = \frac{2}{3}$

$k = 0$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(\frac{2}{3}, 0)$

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.

$\frac{1}{4 \cdot 9}$

Multiply $4$ by $9$ .

$\frac{1}{36}$

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.

$(\frac{2}{3}, \frac{1}{36})$

Step 7

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

$x = \frac{2}{3}$

Step 8

Find the Axis of Symmetry y=9x^2-12x+4

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