Algebra Examples

x^{2} = 4 y

$x^{2} = 4 y$

Step 1

Solve for

y

$y$ .

Tap for more steps...

Step 1.1

Rewrite the equation as

4 y = x^{2}

$4 y = x^{2}$ .

4 y = x^{2}

$4 y = x^{2}$

Step 1.2

Divide each term in

4 y = x^{2}

$4 y = x^{2}$ by

4

$4$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in

4 y = x^{2}

$4 y = x^{2}$ by

4

$4$ .

\frac{4 y}{4} = \frac{x^{2}}{4}

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

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of

4

$4$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

\frac{4 y}{4} = \frac{x^{2}}{4}

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

Step 1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{x^{2}}{4}

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

y = \frac{x^{2}}{4}

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

y = \frac{x^{2}}{4}

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

y = \frac{x^{2}}{4}

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

y = \frac{x^{2}}{4}

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

Step 2

Find the properties of the given parabola.

Tap for more steps...

Step 2.1

Rewrite the equation in vertex form.

Tap for more steps...

Step 2.1.1

Complete the square for

\frac{x^{2}}{4}

$\frac{x^{2}}{4}$ .

Tap for more steps...

Step 2.1.1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = \frac{1}{4}

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

b = 0

$b = 0$

c = 0

$c = 0$

Step 2.1.1.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 2.1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

Tap for more steps...

Step 2.1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{0}{2 (\frac{1}{4})}

$d = \frac{0}{2 (\frac{1}{4})}$

Step 2.1.1.3.2

Simplify the right side.

Tap for more steps...

Step 2.1.1.3.2.1

Cancel the common factor of

0

$0$ and

2

$2$ .

Tap for more steps...

Step 2.1.1.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

d = \frac{2 (0)}{2 (\frac{1}{4})}

$d = \frac{2 (0)}{2 (\frac{1}{4})}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.1.3.2.1.2.1

Cancel the common factor.

d = \frac{2 \cdot 0}{2 (\frac{1}{4})}

$d = \frac{2 \cdot 0}{2 (\frac{1}{4})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

d = \frac{0}{\frac{1}{4}}

$d = \frac{0}{\frac{1}{4}}$

d = \frac{0}{\frac{1}{4}}

$d = \frac{0}{\frac{1}{4}}$

d = \frac{0}{\frac{1}{4}}

$d = \frac{0}{\frac{1}{4}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

d = 0 \cdot 4

$d = 0 \cdot 4$

Step 2.1.1.3.2.3

Multiply

0

$0$ by

4

$4$ .

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

Step 2.1.1.4

Find the value of

e

$e$ using the formula

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

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

Tap for more steps...

Step 2.1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = 0 - \frac{0^{2}}{4 (\frac{1}{4})}

$e = 0 - \frac{0^{2}}{4 (\frac{1}{4})}$

Step 2.1.1.4.2

Simplify the right side.

Tap for more steps...

Step 2.1.1.4.2.1

Simplify each term.

Tap for more steps...

Step 2.1.1.4.2.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

e = 0 - \frac{0}{4 (\frac{1}{4})}

$e = 0 - \frac{0}{4 (\frac{1}{4})}$

Step 2.1.1.4.2.1.2

Combine

4

$4$ and

\frac{1}{4}

$\frac{1}{4}$ .

e = 0 - \frac{0}{\frac{4}{4}}

$e = 0 - \frac{0}{\frac{4}{4}}$

Step 2.1.1.4.2.1.3

Divide

4

$4$ by

4

$4$ .

e = 0 - \frac{0}{1}

$e = 0 - \frac{0}{1}$

Step 2.1.1.4.2.1.4

Divide

0

$0$ by

1

$1$ .

e = 0 - 0

$e = 0 - 0$

Step 2.1.1.4.2.1.5

Multiply

- 1

$- 1$ by

0

$0$ .

e = 0 + 0

$e = 0 + 0$

e = 0 + 0

$e = 0 + 0$

Step 2.1.1.4.2.2

Add

0

$0$ and

0

$0$ .

e = 0

$e = 0$

e = 0

$e = 0$

e = 0

$e = 0$

Step 2.1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

\frac{1}{4} x^{2}

$\frac{1}{4} x^{2}$ .

\frac{1}{4} x^{2}

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

\frac{1}{4} x^{2}

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

Step 2.1.2

Set

y

$y$ equal to the new right side.

y = \frac{1}{4} x^{2}

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

y = \frac{1}{4} x^{2}

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

Step 2.2

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = \frac{1}{4}

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

h = 0

$h = 0$

k = 0

$k = 0$

Step 2.3

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex

(h, k)

$(h, k)$ .

(0, 0)

$(0, 0)$

Step 2.5

Find

p

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

Tap for more steps...

Step 2.5.1

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

\frac{1}{4 a}

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot \frac{1}{4}}

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

Step 2.5.3

Simplify.

Tap for more steps...

Step 2.5.3.1

Combine

4

$4$ and

\frac{1}{4}

$\frac{1}{4}$ .

\frac{1}{\frac{4}{4}}

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

Step 2.5.3.2

Simplify by dividing numbers.

Tap for more steps...

Step 2.5.3.2.1

Divide

4

$4$ by

4

$4$ .

\frac{1}{1}

$\frac{1}{1}$

Step 2.5.3.2.2

Divide

1

$1$ by

1

$1$ .

1

$1$

1

$1$

1

$1$

1

$1$

Step 2.6

Find the focus.

Tap for more steps...

Step 2.6.1

The focus of a parabola can be found by adding

p

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 2.6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(0, 1)

$(0, 1)$

(0, 1)

$(0, 1)$

Step 2.7

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

x = 0

$x = 0$

Step 2.8

Find the directrix.

Tap for more steps...

Step 2.8.1

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

p

$p$ from the y-coordinate

k

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

y = k - p

$y = k - p$

Step 2.8.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = - 1

$y = - 1$

y = - 1

$y = - 1$

Step 2.9

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

Direction: Opens Up

Vertex:

(0, 0)

$(0, 0)$

Focus:

(0, 1)

$(0, 1)$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = - 1

$y = - 1$

Direction: Opens Up

Vertex:

(0, 0)

$(0, 0)$

Focus:

(0, 1)

$(0, 1)$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = - 1

$y = - 1$

Step 3

Select a few

x

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

y

$y$ values. The

x

$x$ values should be selected around the vertex.

Tap for more steps...

Step 3.1

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

f (- 2) = \frac{4}{4}

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

Step 3.2.2

Divide

4

$4$ by

4

$4$ .

f (- 2) = 1

$f (- 2) = 1$

Step 3.2.3

The final answer is

1

$1$ .

1

$1$

1

$1$

Step 3.3

The

y

$y$ value at

x = - 2

$x = - 2$ is

1

$1$ .

y = 1

$y = 1$

Step 3.4

Replace the variable

x

$x$ with

- 1

$- 1$ in the expression.

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

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

Step 3.5

Simplify the result.

Tap for more steps...

Step 3.5.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

f (- 1) = \frac{1}{4}

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

Step 3.5.2

The final answer is

\frac{1}{4}

$\frac{1}{4}$ .

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

Step 3.6

The

y

$y$ value at

x = - 1

$x = - 1$ is

\frac{1}{4}

$\frac{1}{4}$ .

y = \frac{1}{4}

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

Step 3.7

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 3.8

Simplify the result.

Tap for more steps...

Step 3.8.1

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 3.8.2

Divide

4

$4$ by

4

$4$ .

f (2) = 1

$f (2) = 1$

Step 3.8.3

The final answer is

1

$1$ .

1

$1$

1

$1$

Step 3.9

The

y

$y$ value at

x = 2

$x = 2$ is

1

$1$ .

y = 1

$y = 1$

Step 3.10

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 3.11

Simplify the result.

Tap for more steps...

Step 3.11.1

One to any power is one.

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

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

Step 3.11.2

The final answer is

\frac{1}{4}

$\frac{1}{4}$ .

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

Step 3.12

The

y

$y$ value at

x = 1

$x = 1$ is

\frac{1}{4}

$\frac{1}{4}$ .

y = \frac{1}{4}

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

Step 3.13

Graph the parabola using its properties and the selected points.

\begin{matrix} x & y - 2 & 1 - 1 & \frac{1}{4} 0 & 0 1 & \frac{1}{4} 2 & 1 \end{matrix}

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

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex:

$(0, 0)$

Focus:

$(0, 1)$

Axis of Symmetry:

$x = 0$

Directrix:

$y = - 1$

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

Step 5