Find the Axis of Symmetry y=4x^2+6x-5

$y = 4 x^{2} + 6 x - 5$

Step 1

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} + 6 x - 5$ .

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

$a = 4$

$b = 6$

$c = - 5$

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

$d = \frac{6}{2 \cdot 4}$

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

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

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

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot 4}$

Rewrite the expression.

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

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

$e = - 5 - \frac{6^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

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

$e = - 5 - \frac{36}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = - 5 - \frac{36}{16}$

Cancel the common factor of $36$ and $16$ .

Factor $4$ out of $36$ .

$e = - 5 - \frac{4 (9)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 5$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 5$ by $4$ .

$e = \frac{- 20 - 9}{4}$

Subtract $9$ from $- 20$ .

$e = \frac{- 29}{4}$

Move the negative in front of the fraction.

$e = - \frac{29}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{3}{4})}^{2} - \frac{29}{4}$ .

$4 {(x + \frac{3}{4})}^{2} - \frac{29}{4}$

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{3}{4})}^{2} - \frac{29}{4}$

Step 2

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

$a = 4$

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

$k = - \frac{29}{4}$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- \frac{3}{4}, - \frac{29}{4})$

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.

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

Multiply $4$ by $4$ .

$\frac{1}{16}$

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.

$(- \frac{3}{4}, - \frac{115}{16})$

Step 7

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

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

Step 8

Find the Axis of Symmetry y=4x^2+6x-5

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