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

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

Step 1

Rewrite the equation in vertex form.

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

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

Complete the square for $\frac{x^{2}}{2} + 3 x + \frac{1}{2}$ .

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

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

$b = 3$

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

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

$d = \frac{3}{2 (\frac{1}{2})}$

Simplify the right side.

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

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

Divide $2$ by $2$ .

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

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

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = \frac{1}{2} - \frac{9}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$e = \frac{1}{2} - \frac{9}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = \frac{1}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$e = \frac{1 - 9}{2}$

Subtract $9$ from $1$ .

$e = \frac{- 8}{2}$

Divide $- 8$ by $2$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x + 3)}^{2} - 4$ .

$\frac{1}{2} {(x + 3)}^{2} - 4$

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 3)}^{2} - 4$

Step 2

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

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

$h = - 3$

$k = - 4$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 3, - 4)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{2}$ .

$\frac{1}{\frac{4}{2}}$

Divide $4$ by $2$ .

$\frac{1}{2}$

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.

$(- 3, - \frac{7}{2})$

Step 7

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

$x = - 3$

Step 8

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

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