Find the Axis of Symmetry y-4=1/2(x+1)^2

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

Step 1

Isolate $y$ to the left side of the equation.

Simplify $\frac{1}{2} {(x + 1)}^{2}$ .

Rewrite.

$y - 4 = 0 + 0 + \frac{1}{2} {(x + 1)}^{2}$

Simplify by adding zeros.

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

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

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

Add $4$ to both sides of the equation.

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

Reorder terms.

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

Step 2

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

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

$h = - 1$

$k = 4$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 1, 4)$

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.

$(- 1, \frac{9}{2})$

Step 7

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

$x = - 1$

Step 8

Find the Axis of Symmetry y-4=1/2(x+1)^2

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