Algebra Examples

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

Step 1

Find the properties of the given parabola.

Tap for more steps...

Step 1.1

Isolate

$y$ to the left side of the equation.

Tap for more steps...

Step 1.1.1

Divide each term in

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

$2$ and simplify.

Tap for more steps...

Step 1.1.1.1

Divide each term in

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

$2$ .

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

Step 1.1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.1.2.1

Cancel the common factor of

$2$ .

Tap for more steps...

Step 1.1.1.2.1.1

Cancel the common factor.

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

Step 1.1.1.2.1.2

Divide

$y$ by

$1$ .

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

Step 1.1.2

Reorder terms.

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

Step 1.2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

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

$h = 3$

$k = 0$

Step 1.3

Since the value of

$a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex

$(h, k)$ .

$(3, 0)$

Step 1.5

Find

$p$ , the distance from the vertex to the focus.

Tap for more steps...

Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of

$a$ into the formula.

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

Step 1.5.3

Simplify.

Tap for more steps...

Step 1.5.3.1

Combine

$4$ and

$\frac{1}{2}$ .

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

Step 1.5.3.2

Divide

$4$ by

$2$ .

$\frac{1}{2}$

Step 1.6

Find the focus.

Tap for more steps...

Step 1.6.1

The focus of a parabola can be found by adding

$p$ to the y-coordinate

$k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(3, \frac{1}{2})$

Step 1.7

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

$x = 3$

Step 1.8

Find the directrix.

Tap for more steps...

Step 1.8.1

The directrix of a parabola is the horizontal line found by subtracting

$p$ from the y-coordinate

$k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.8.2

Substitute the known values of

$p$ and

$k$ into the formula and simplify.

$y = - \frac{1}{2}$

Step 1.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex:

$(3, 0)$

Focus:

$(3, \frac{1}{2})$

Axis of Symmetry:

$x = 3$

Directrix:

$y = - \frac{1}{2}$

Direction: Opens Up

Vertex:

$(3, 0)$

Focus:

$(3, \frac{1}{2})$

Axis of Symmetry:

$x = 3$

Directrix:

$y = - \frac{1}{2}$

Step 2

Select a few

$x$ values, and plug them into the equation to find the corresponding

$y$ values. The

$x$ values should be selected around the vertex.

Tap for more steps...

Step 2.1

Replace the variable

$x$ with

$1$ in the expression.

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

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.1

Subtract

$3$ from

$1$ .

$f (1) = \frac{{(- 2)}^{2}}{2}$

Step 2.2.1.2

Raise

$- 2$ to the power of

$2$ .

$f (1) = \frac{4}{2}$

Step 2.2.2

Divide

$4$ by

$2$ .

$f (1) = 2$

Step 2.2.3

The final answer is

$2$ .

$2$

Step 2.3

The

$y$ value at

$x = 1$ is

$2$ .

$y = 2$

Step 2.4

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = \frac{{((2) - 3)}^{2}}{2}$

Step 2.5

Simplify the result.

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.1.1

Subtract

$3$ from

$2$ .

$f (2) = \frac{{(- 1)}^{2}}{2}$

Step 2.5.1.2

Raise

$- 1$ to the power of

$2$ .

$f (2) = \frac{1}{2}$

Step 2.5.2

The final answer is

$\frac{1}{2}$ .

$\frac{1}{2}$

Step 2.6

The

$y$ value at

$x = 2$ is

$\frac{1}{2}$ .

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

Step 2.7

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = \frac{{((5) - 3)}^{2}}{2}$

Step 2.8

Simplify the result.

Tap for more steps...

Step 2.8.1

Simplify the numerator.

Tap for more steps...

Step 2.8.1.1

Subtract

$3$ from

$5$ .

$f (5) = \frac{2^{2}}{2}$

Step 2.8.1.2

Raise

$2$ to the power of

$2$ .

$f (5) = \frac{4}{2}$

Step 2.8.2

Divide

$4$ by

$2$ .

$f (5) = 2$

Step 2.8.3

The final answer is

$2$ .

$2$

Step 2.9

The

$y$ value at

$x = 5$ is

$2$ .

$y = 2$

Step 2.10

Replace the variable

$x$ with

$4$ in the expression.

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

Step 2.11

Simplify the result.

Tap for more steps...

Step 2.11.1

Simplify the numerator.

Tap for more steps...

Step 2.11.1.1

Subtract

$3$ from

$4$ .

$f (4) = \frac{1^{2}}{2}$

Step 2.11.1.2

One to any power is one.

$f (4) = \frac{1}{2}$

Step 2.11.2

The final answer is

$\frac{1}{2}$ .

$\frac{1}{2}$

Step 2.12

The

$y$ value at

$x = 4$ is

$\frac{1}{2}$ .

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

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 2 \\ 2 & \frac{1}{2} \\ 3 & 0 \\ 4 & \frac{1}{2} \\ 5 & 2 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(3, 0)$

Focus:

$(3, \frac{1}{2})$

Axis of Symmetry:

$x = 3$

Directrix:

$y = - \frac{1}{2}$

$\begin{array}{c} x & y \\ 1 & 2 \\ 2 & \frac{1}{2} \\ 3 & 0 \\ 4 & \frac{1}{2} \\ 5 & 2 \end{array}$

Step 4