Algebra Examples

y = - 0.25 x^{2} + 5

$y = - 0.25 x^{2} + 5$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

- 0.25 x^{2} + 5

$- 0.25 x^{2} + 5$ .

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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 = - 0.25

$a = - 0.25$

b = 0

$b = 0$

c = 5

$c = 5$

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 - 0.25}

$d = \frac{0}{2 \cdot - 0.25}$

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 - 0.25}

$d = \frac{2 (0)}{2 \cdot - 0.25}$

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 - 0.25

$2 \cdot - 0.25$ .

d = \frac{2 (0)}{2 (- 0.25)}

$d = \frac{2 (0)}{2 (- 0.25)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

d = \frac{2 \cdot 0}{2 \cdot - 0.25}

$d = \frac{2 \cdot 0}{2 \cdot - 0.25}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

d = \frac{0}{- 0.25}

$d = \frac{0}{- 0.25}$

d = \frac{0}{- 0.25}

$d = \frac{0}{- 0.25}$

d = \frac{0}{- 0.25}

$d = \frac{0}{- 0.25}$

Step 1.1.3.2.2

Divide

0

$0$ by

- 0.25

$- 0.25$ .

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 = 5 - \frac{0^{2}}{4 \cdot - 0.25}

$e = 5 - \frac{0^{2}}{4 \cdot - 0.25}$

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 = 5 - \frac{0}{4 \cdot - 0.25}

$e = 5 - \frac{0}{4 \cdot - 0.25}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

- 0.25

$- 0.25$ .

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

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

Step 1.1.4.2.1.3

Divide

0

$0$ by

- 1

$- 1$ .

e = 5 - 0

$e = 5 - 0$

Step 1.1.4.2.1.4

Multiply

- 1

$- 1$ by

0

$0$ .

e = 5 + 0

$e = 5 + 0$

e = 5 + 0

$e = 5 + 0$

Step 1.1.4.2.2

Add

5

$5$ and

0

$0$ .

e = 5

$e = 5$

e = 5

$e = 5$

e = 5

$e = 5$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

- 0.25 {(x + 0)}^{2} + 5

$- 0.25 {(x + 0)}^{2} + 5$ .

- 0.25 {(x + 0)}^{2} + 5

$- 0.25 {(x + 0)}^{2} + 5$

- 0.25 {(x + 0)}^{2} + 5

$- 0.25 {(x + 0)}^{2} + 5$

Step 1.2

Set

y

$y$ equal to the new right side.

y = - 0.25 {(x + 0)}^{2} + 5

$y = - 0.25 {(x + 0)}^{2} + 5$

y = - 0.25 {(x + 0)}^{2} + 5

$y = - 0.25 {(x + 0)}^{2} + 5$

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 = - 0.25

$a = - 0.25$

h = 0

$h = 0$

k = 5

$k = 5$

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(0, 5)

$(0, 5)$

Step 4

Find

p

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

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Step 4.1

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 4.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot - 0.25}

$\frac{1}{4 \cdot - 0.25}$

Step 4.3

Simplify.

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Step 4.3.1

Multiply

4

$4$ by

- 0.25

$- 0.25$ .

\frac{1}{- 1}

$\frac{1}{- 1}$

Step 4.3.2

Divide

1

$1$ by

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

- 1

$- 1$

Step 5

Find the focus.

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Step 5.1

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 5.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(0, 4)

$(0, 4)$

(0, 4)

$(0, 4)$

Step 6