Precalculus Examples

x = 7 y^{2}

$x = 7 y^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

7 y^{2}

$7 y^{2}$ .

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Step 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 = 7

$a = 7$

b = 0

$b = 0$

c = 0

$c = 0$

Step 1.1.2

Consider the vertex form of a parabola.

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

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

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.1.3.1

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 \cdot 7}

$d = \frac{0}{2 \cdot 7}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 1.1.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

d = \frac{2 (0)}{2 \cdot 7}

$d = \frac{2 (0)}{2 \cdot 7}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot 7

$2 \cdot 7$ .

d = \frac{2 (0)}{2 (7)}

$d = \frac{2 (0)}{2 (7)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

d = \frac{2 \cdot 0}{2 \cdot 7}

$d = \frac{2 \cdot 0}{2 \cdot 7}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

d = \frac{0}{7}

$d = \frac{0}{7}$

d = \frac{0}{7}

$d = \frac{0}{7}$

d = \frac{0}{7}

$d = \frac{0}{7}$

Step 1.1.3.2.2

Cancel the common factor of

0

$0$ and

7

$7$ .

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Step 1.1.3.2.2.1

Factor

7

$7$ out of

0

$0$ .

d = \frac{7 (0)}{7}

$d = \frac{7 (0)}{7}$

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor

7

$7$ out of

7

$7$ .

d = \frac{7 \cdot 0}{7 \cdot 1}

$d = \frac{7 \cdot 0}{7 \cdot 1}$

Step 1.1.3.2.2.2.2

Cancel the common factor.

d = \frac{7 \cdot 0}{7 \cdot 1}

$d = \frac{7 \cdot 0}{7 \cdot 1}$

Step 1.1.3.2.2.2.3

Rewrite the expression.

d = \frac{0}{1}

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

Step 1.1.3.2.2.2.4

Divide

0

$0$ by

1

$1$ .

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

Step 1.1.4

Find the value of

e

$e$ using the formula

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

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

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Step 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 \cdot 7}

$e = 0 - \frac{0^{2}}{4 \cdot 7}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

e = 0 - \frac{0}{4 \cdot 7}

$e = 0 - \frac{0}{4 \cdot 7}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

7

$7$ .

e = 0 - \frac{0}{28}

$e = 0 - \frac{0}{28}$

Step 1.1.4.2.1.3

Divide

0

$0$ by

28

$28$ .

e = 0 - 0

$e = 0 - 0$

Step 1.1.4.2.1.4

Multiply

- 1

$- 1$ by

0

$0$ .

e = 0 + 0

$e = 0 + 0$

e = 0 + 0

$e = 0 + 0$

Step 1.1.4.2.2

Add

0

$0$ and

0

$0$ .

e = 0

$e = 0$

e = 0

$e = 0$

e = 0

$e = 0$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

7 y^{2}

$7 y^{2}$ .

7 y^{2}

$7 y^{2}$

7 y^{2}

$7 y^{2}$

Step 1.2

Set

x

$x$ equal to the new right side.

x = 7 y^{2}

$x = 7 y^{2}$

x = 7 y^{2}

$x = 7 y^{2}$

Step 2

Use the vertex form,

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

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 7

$a = 7$

h = 0

$h = 0$

k = 0

$k = 0$

Step 3

Since the value of

a

$a$ is positive, the parabola opens right.

Opens Right

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 7}

$\frac{1}{4 \cdot 7}$

Step 5.3

Multiply

4

$4$ by

7

$7$ .

\frac{1}{28}

$\frac{1}{28}$

\frac{1}{28}

$\frac{1}{28}$

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 x-coordinate

h

$h$ if the parabola opens left or right.

(h + p, k)

$(h + p, k)$

Step 6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(\frac{1}{28}, 0)

$(\frac{1}{28}, 0)$

(\frac{1}{28}, 0)

$(\frac{1}{28}, 0)$

Step 7

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

y = 0

$y = 0$

Step 8

Find the directrix.

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Step 8.1

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

p

$p$ from the x-coordinate

h

$h$ of the vertex if the parabola opens left or right.

x = h - p

$x = h - p$

Step 8.2

Substitute the known values of

p

$p$ and

h

$h$ into the formula and simplify.

x = - \frac{1}{28}

$x = - \frac{1}{28}$

x = - \frac{1}{28}

$x = - \frac{1}{28}$

Step 9

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

Direction: Opens Right

Vertex:

(0, 0)

$(0, 0)$

Focus:

(\frac{1}{28}, 0)

$(\frac{1}{28}, 0)$

Axis of Symmetry:

y = 0

$y = 0$

Directrix:

x = - \frac{1}{28}

$x = - \frac{1}{28}$

Step 10