Precalculus Examples

x^{2} = 20 y

$x^{2} = 20 y$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate

y

$y$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as

20 y = x^{2}

$20 y = x^{2}$ .

20 y = x^{2}

$20 y = x^{2}$

Step 1.1.2

Divide each term in

20 y = x^{2}

$20 y = x^{2}$ by

20

$20$ and simplify.

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Step 1.1.2.1

Divide each term in

20 y = x^{2}

$20 y = x^{2}$ by

20

$20$ .

\frac{20 y}{20} = \frac{x^{2}}{20}

$\frac{20 y}{20} = \frac{x^{2}}{20}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of

20

$20$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

\frac{20 y}{20} = \frac{x^{2}}{20}

$\frac{20 y}{20} = \frac{x^{2}}{20}$

Step 1.1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{x^{2}}{20}

$y = \frac{x^{2}}{20}$

y = \frac{x^{2}}{20}

$y = \frac{x^{2}}{20}$

y = \frac{x^{2}}{20}

$y = \frac{x^{2}}{20}$

y = \frac{x^{2}}{20}

$y = \frac{x^{2}}{20}$

y = \frac{x^{2}}{20}

$y = \frac{x^{2}}{20}$

Step 1.2

Complete the square for

\frac{x^{2}}{20}

$\frac{x^{2}}{20}$ .

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Step 1.2.1

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}{20}

$a = \frac{1}{20}$

b = 0

$b = 0$

c = 0

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

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

Step 1.2.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.2.3.1

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}{20})}

$d = \frac{0}{2 (\frac{1}{20})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 1.2.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

d = \frac{2 (0)}{2 (\frac{1}{20})}

$d = \frac{2 (0)}{2 (\frac{1}{20})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

d = \frac{2 \cdot 0}{2 (\frac{1}{20})}

$d = \frac{2 \cdot 0}{2 (\frac{1}{20})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

d = \frac{0}{\frac{1}{20}}

$d = \frac{0}{\frac{1}{20}}$

d = \frac{0}{\frac{1}{20}}

$d = \frac{0}{\frac{1}{20}}$

d = \frac{0}{\frac{1}{20}}

$d = \frac{0}{\frac{1}{20}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

d = 0 \cdot 20

$d = 0 \cdot 20$

Step 1.2.3.2.3

Multiply

0

$0$ by

20

$20$ .

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

Step 1.2.4

Find the value of

e

$e$ using the formula

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

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

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Step 1.2.4.1

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}{20})}

$e = 0 - \frac{0^{2}}{4 (\frac{1}{20})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

e = 0 - \frac{0}{4 (\frac{1}{20})}

$e = 0 - \frac{0}{4 (\frac{1}{20})}$

Step 1.2.4.2.1.2

Combine

4

$4$ and

\frac{1}{20}

$\frac{1}{20}$ .

e = 0 - \frac{0}{\frac{4}{20}}

$e = 0 - \frac{0}{\frac{4}{20}}$

Step 1.2.4.2.1.3

Cancel the common factor of

4

$4$ and

20

$20$ .

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Step 1.2.4.2.1.3.1

Factor

4

$4$ out of

4

$4$ .

e = 0 - \frac{0}{\frac{4 (1)}{20}}

$e = 0 - \frac{0}{\frac{4 (1)}{20}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor

4

$4$ out of

20

$20$ .

e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 5}}

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 5}}

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

e = 0 - \frac{0}{\frac{1}{5}}

$e = 0 - \frac{0}{\frac{1}{5}}$

e = 0 - \frac{0}{\frac{1}{5}}

$e = 0 - \frac{0}{\frac{1}{5}}$

e = 0 - \frac{0}{\frac{1}{5}}

$e = 0 - \frac{0}{\frac{1}{5}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

e = 0 - (0 \cdot 5)

$e = 0 - (0 \cdot 5)$

Step 1.2.4.2.1.5

Multiply

- (0 \cdot 5)

$- (0 \cdot 5)$ .

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Step 1.2.4.2.1.5.1

Multiply

0

$0$ by

5

$5$ .

e = 0 - 0

$e = 0 - 0$

Step 1.2.4.2.1.5.2

Multiply

- 1

$- 1$ by

0

$0$ .

e = 0 + 0

$e = 0 + 0$

e = 0 + 0

$e = 0 + 0$

e = 0 + 0

$e = 0 + 0$

Step 1.2.4.2.2

Add

0

$0$ and

0

$0$ .

e = 0

$e = 0$

e = 0

$e = 0$

e = 0

$e = 0$

Step 1.2.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

\frac{1}{20} x^{2}

$\frac{1}{20} x^{2}$ .

\frac{1}{20} x^{2}

$\frac{1}{20} x^{2}$

\frac{1}{20} x^{2}

$\frac{1}{20} x^{2}$

Step 1.3

Set

y

$y$ equal to the new right side.

y = \frac{1}{20} x^{2}

$y = \frac{1}{20} x^{2}$

y = \frac{1}{20} x^{2}

$y = \frac{1}{20} x^{2}$

Step 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}{20}

$a = \frac{1}{20}$

h = 0

$h = 0$

k = 0

$k = 0$

Step 3

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 4

Find the vertex

(h, k)

$(h, k)$ .

(0, 0)

$(0, 0)$

Step 5

Find

p

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

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Step 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 5.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot \frac{1}{20}}

$\frac{1}{4 \cdot \frac{1}{20}}$

Step 5.3

Simplify.

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Step 5.3.1

Combine

4

$4$ and

\frac{1}{20}

$\frac{1}{20}$ .

\frac{1}{\frac{4}{20}}

$\frac{1}{\frac{4}{20}}$

Step 5.3.2

Cancel the common factor of

4

$4$ and

20

$20$ .

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Step 5.3.2.1

Factor

4

$4$ out of

4

$4$ .

\frac{1}{\frac{4 (1)}{20}}

$\frac{1}{\frac{4 (1)}{20}}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor

4

$4$ out of

20

$20$ .

\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 5.3.2.2.2

Cancel the common factor.

\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 5.3.2.2.3

Rewrite the expression.

\frac{1}{\frac{1}{5}}

$\frac{1}{\frac{1}{5}}$

\frac{1}{\frac{1}{5}}

$\frac{1}{\frac{1}{5}}$

\frac{1}{\frac{1}{5}}

$\frac{1}{\frac{1}{5}}$

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

1 \cdot 5

$1 \cdot 5$

Step 5.3.4

Multiply

5

$5$ by

1

$1$ .

5

$5$

5

$5$

5

$5$

Step 6

Find the focus.

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Step 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 6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(0, 5)

$(0, 5)$

(0, 5)

$(0, 5)$

Step 7

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

x = 0

$x = 0$

Step 8

Find the directrix.

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Step 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 8.2

Substitute the known values of

p

$p$ and

k

$k$ into the formula and simplify.

y = - 5

$y = - 5$

y = - 5

$y = - 5$

Step 9

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

Direction: Opens Up

Vertex:

(0, 0)

$(0, 0)$

Focus:

(0, 5)

$(0, 5)$

Axis of Symmetry:

x = 0

$x = 0$

Directrix:

y = - 5

$y = - 5$

Step 10