Find the Axis of Symmetry -4x^2+2x+3

$- 4 x^{2} + 2 x + 3$

Step 1

Write $- 4 x^{2} + 2 x + 3$ as an equation.

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

Step 2

Rewrite the equation in vertex form.

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

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

$a = - 4$

$b = 2$

$c = 3$

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

$d = \frac{2}{2 \cdot - 4}$

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot - 4}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $- 4$ .

$e = 3 - \frac{4}{- 16}$

Cancel the common factor of $4$ and $- 16$ .

Factor $4$ out of $4$ .

$e = 3 - \frac{4 (1)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{1}{4}$ .

Multiply $- 1$ by $- 1$ .

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

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

$e = 3 + \frac{1}{4}$

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

$e = 3 \cdot \frac{4}{4} + \frac{1}{4}$

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

$e = \frac{3 \cdot 4}{4} + \frac{1}{4}$

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 4 + 1}{4}$

Simplify the numerator.

Multiply $3$ by $4$ .

$e = \frac{12 + 1}{4}$

Add $12$ and $1$ .

$e = \frac{13}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x - \frac{1}{4})}^{2} + \frac{13}{4}$ .

$- 4 {(x - \frac{1}{4})}^{2} + \frac{13}{4}$

Set $y$ equal to the new right side.

$y = - 4 {(x - \frac{1}{4})}^{2} + \frac{13}{4}$

Step 3

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

$a = - 4$

$h = \frac{1}{4}$

$k = \frac{13}{4}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(\frac{1}{4}, \frac{13}{4})$

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.

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

Simplify.

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Move the negative in front of the fraction.

$- \frac{1}{16}$

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.

$(\frac{1}{4}, \frac{51}{16})$

Step 8

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

$x = \frac{1}{4}$

Step 9

Find the Axis of Symmetry -4x^2+2x+3

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