Find the Axis of Symmetry y=4x^2+32x+61

$y = 4 x^{2} + 32 x + 61$

Step 1

Rewrite the equation in vertex form.

Complete the square for $4 x^{2} + 32 x + 61$ .

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

$a = 4$

$b = 32$

$c = 61$

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

$d = \frac{32}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $32$ and $2$ .

Factor $2$ out of $32$ .

$d = \frac{2 \cdot 16}{2 \cdot 4}$

Cancel the common factors.

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

$d = \frac{2 \cdot 16}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot 16}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{16}{4}$

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

Factor $4$ out of $16$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot 4}{4 (1)}$

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$d = 4$

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

$e = 61 - \frac{32^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $32$ to the power of $2$ .

$e = 61 - \frac{1024}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 61 - \frac{1024}{16}$

Divide $1024$ by $16$ .

$e = 61 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 61 - 64$

Subtract $64$ from $61$ .

$e = - 3$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

$a = 4$

$h = - 4$

$k = - 3$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 4, - 3)$

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.

$(- 4, - \frac{47}{16})$

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=4x^2+32x+61

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