Find the Axis of Symmetry y=4x^2+1/2x+3

$y = 4 x^{2} + \frac{1}{2} x + 3$

Step 1

Rewrite the equation in vertex form.

Combine $\frac{1}{2}$ and $x$ .

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

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

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

$a = 4$

$b = \frac{1}{2}$

$c = 3$

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

$d = \frac{\frac{1}{2}}{2 \cdot 4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} \cdot \frac{1}{2 \cdot 4}$

Multiply $2$ by $4$ .

$d = \frac{1}{2} \cdot \frac{1}{8}$

Multiply $\frac{1}{2} \cdot \frac{1}{8}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$d = \frac{1}{2 \cdot 8}$

Multiply $2$ by $8$ .

$d = \frac{1}{16}$

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

$e = 3 - \frac{{(\frac{1}{2})}^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{2}$ .

$e = 3 - \frac{\frac{1^{2}}{2^{2}}}{4 \cdot 4}$

One to any power is one.

$e = 3 - \frac{\frac{1}{2^{2}}}{4 \cdot 4}$

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

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

Multiply $4$ by $4$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{1}{4} \cdot \frac{1}{16}$ .

Multiply $\frac{1}{4}$ by $\frac{1}{16}$ .

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

Multiply $4$ by $16$ .

$e = 3 - \frac{1}{64}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$e = 3 \cdot \frac{64}{64} - \frac{1}{64}$

Combine $3$ and $\frac{64}{64}$ .

$e = \frac{3 \cdot 64}{64} - \frac{1}{64}$

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 64 - 1}{64}$

Simplify the numerator.

Multiply $3$ by $64$ .

$e = \frac{192 - 1}{64}$

Subtract $1$ from $192$ .

$e = \frac{191}{64}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{1}{16})}^{2} + \frac{191}{64}$ .

$4 {(x + \frac{1}{16})}^{2} + \frac{191}{64}$

Set $y$ equal to the new right side.

$y = 4 {(x + \frac{1}{16})}^{2} + \frac{191}{64}$

Step 2

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

$a = 4$

$h = - \frac{1}{16}$

$k = \frac{191}{64}$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- \frac{1}{16}, \frac{191}{64})$

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.

$(- \frac{1}{16}, \frac{195}{64})$

Step 7

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

$x = - \frac{1}{16}$

Step 8

Find the Axis of Symmetry y=4x^2+1/2x+3

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