Algebra Examples

f (x) = - 3 x^{2} + 12 x - 6

$f (x) = - 3 x^{2} + 12 x - 6$

Step 1

Write

f (x) = - 3 x^{2} + 12 x - 6

$f (x) = - 3 x^{2} + 12 x - 6$ as an equation.

y = - 3 x^{2} + 12 x - 6

$y = - 3 x^{2} + 12 x - 6$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for

- 3 x^{2} + 12 x - 6

$- 3 x^{2} + 12 x - 6$ .

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Step 2.1.1

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 = - 3

$a = - 3$

b = 12

$b = 12$

c = - 6

$c = - 6$

Step 2.1.2

Consider the vertex form of a parabola.

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

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

Step 2.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{12}{2 \cdot - 3}

$d = \frac{12}{2 \cdot - 3}$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Cancel the common factor of

12

$12$ and

2

$2$ .

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Step 2.1.3.2.1.1

Factor

2

$2$ out of

12

$12$ .

d = \frac{2 \cdot 6}{2 \cdot - 3}

$d = \frac{2 \cdot 6}{2 \cdot - 3}$

Step 2.1.3.2.1.2

Cancel the common factors.

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Step 2.1.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot - 3

$2 \cdot - 3$ .

d = \frac{2 \cdot 6}{2 (- 3)}

$d = \frac{2 \cdot 6}{2 (- 3)}$

Step 2.1.3.2.1.2.2

Cancel the common factor.

d = \frac{2 \cdot 6}{2 \cdot - 3}

$d = \frac{2 \cdot 6}{2 \cdot - 3}$

Step 2.1.3.2.1.2.3

Rewrite the expression.

d = \frac{6}{- 3}

$d = \frac{6}{- 3}$

d = \frac{6}{- 3}

$d = \frac{6}{- 3}$

d = \frac{6}{- 3}

$d = \frac{6}{- 3}$

Step 2.1.3.2.2

Cancel the common factor of

6

$6$ and

- 3

$- 3$ .

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Step 2.1.3.2.2.1

Factor

3

$3$ out of

6

$6$ .

d = \frac{3 (2)}{- 3}

$d = \frac{3 (2)}{- 3}$

Step 2.1.3.2.2.2

Move the negative one from the denominator of

\frac{2}{- 1}

$\frac{2}{- 1}$ .

d = - 1 \cdot 2

$d = - 1 \cdot 2$

d = - 1 \cdot 2

$d = - 1 \cdot 2$

Step 2.1.3.2.3

Multiply

- 1

$- 1$ by

2

$2$ .

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

Step 2.1.4

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 2.1.4.1

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 = - 6 - \frac{12^{2}}{4 \cdot - 3}

$e = - 6 - \frac{12^{2}}{4 \cdot - 3}$

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

Raise

12

$12$ to the power of

2

$2$ .

e = - 6 - \frac{144}{4 \cdot - 3}

$e = - 6 - \frac{144}{4 \cdot - 3}$

Step 2.1.4.2.1.2

Multiply

4

$4$ by

- 3

$- 3$ .

e = - 6 - \frac{144}{- 12}

$e = - 6 - \frac{144}{- 12}$

Step 2.1.4.2.1.3

Divide

144

$144$ by

- 12

$- 12$ .

e = - 6 - - 12

$e = - 6 - - 12$

Step 2.1.4.2.1.4

Multiply

- 1

$- 1$ by

- 12

$- 12$ .

e = - 6 + 12

$e = - 6 + 12$

e = - 6 + 12

$e = - 6 + 12$

Step 2.1.4.2.2

Add

- 6

$- 6$ and

12

$12$ .

e = 6

$e = 6$

e = 6

$e = 6$

e = 6

$e = 6$

Step 2.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

- 3 {(x - 2)}^{2} + 6

$- 3 {(x - 2)}^{2} + 6$ .

- 3 {(x - 2)}^{2} + 6

$- 3 {(x - 2)}^{2} + 6$

- 3 {(x - 2)}^{2} + 6

$- 3 {(x - 2)}^{2} + 6$

Step 2.2

Set

y

$y$ equal to the new right side.

y = - 3 {(x - 2)}^{2} + 6

$y = - 3 {(x - 2)}^{2} + 6$

y = - 3 {(x - 2)}^{2} + 6

$y = - 3 {(x - 2)}^{2} + 6$

Step 3

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 = - 3

$a = - 3$

h = 2

$h = 2$

k = 6

$k = 6$

Step 4

Since the value of

a

$a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex

(h, k)

$(h, k)$ .

(2, 6)

$(2, 6)$

Step 6

Find

p

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

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Step 6.1

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 6.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot - 3}

$\frac{1}{4 \cdot - 3}$

Step 6.3

Simplify.

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Step 6.3.1

Multiply

4

$4$ by

- 3

$- 3$ .

\frac{1}{- 12}

$\frac{1}{- 12}$

Step 6.3.2

Move the negative in front of the fraction.

- \frac{1}{12}

$- \frac{1}{12}$

- \frac{1}{12}

$- \frac{1}{12}$

- \frac{1}{12}

$- \frac{1}{12}$

Step 7

Find the focus.

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Step 7.1

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 7.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(2, \frac{71}{12})

$(2, \frac{71}{12})$

(2, \frac{71}{12})

$(2, \frac{71}{12})$

Step 8

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

x = 2

$x = 2$

Step 9