Precalculus Examples

f (x) = 3 x^{2} - 12 x + 1

$f (x) = 3 x^{2} - 12 x + 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

3 x^{2} - 12 x + 1

$3 x^{2} - 12 x + 1$ .

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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 = 3

$a = 3$

b = - 12

$b = - 12$

c = 1

$c = 1$

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{- 12}{2 \cdot 3}

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of

- 12

$- 12$ and

2

$2$ .

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Step 1.1.3.2.1.1

Factor

2

$2$ out of

- 12

$- 12$ .

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

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

Step 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 3

$2 \cdot 3$ .

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

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

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

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

Step 1.1.3.2.1.2.3

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 1.1.3.2.2

Cancel the common factor of

- 6

$- 6$ and

3

$3$ .

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Step 1.1.3.2.2.1

Factor

3

$3$ out of

- 6

$- 6$ .

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

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

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor

3

$3$ out of

3

$3$ .

d = \frac{3 \cdot - 2}{3 (1)}

$d = \frac{3 \cdot - 2}{3 (1)}$

Step 1.1.3.2.2.2.2

Cancel the common factor.

d = \frac{3 \cdot - 2}{3 \cdot 1}

$d = \frac{3 \cdot - 2}{3 \cdot 1}$

Step 1.1.3.2.2.2.3

Rewrite the expression.

d = \frac{- 2}{1}

$d = \frac{- 2}{1}$

Step 1.1.3.2.2.2.4

Divide

- 2

$- 2$ by

1

$1$ .

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

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 = 1 - \frac{{(- 12)}^{2}}{4 \cdot 3}

$e = 1 - \frac{{(- 12)}^{2}}{4 \cdot 3}$

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

Raise

- 12

$- 12$ to the power of

2

$2$ .

e = 1 - \frac{144}{4 \cdot 3}

$e = 1 - \frac{144}{4 \cdot 3}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

3

$3$ .

e = 1 - \frac{144}{12}

$e = 1 - \frac{144}{12}$

Step 1.1.4.2.1.3

Divide

144

$144$ by

12

$12$ .

e = 1 - 1 \cdot 12

$e = 1 - 1 \cdot 12$

Step 1.1.4.2.1.4

Multiply

- 1

$- 1$ by

12

$12$ .

e = 1 - 12

$e = 1 - 12$

e = 1 - 12

$e = 1 - 12$

Step 1.1.4.2.2

Subtract

12

$12$ from

1

$1$ .

e = - 11

$e = - 11$

e = - 11

$e = - 11$

e = - 11

$e = - 11$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

3 {(x - 2)}^{2} - 11

$3 {(x - 2)}^{2} - 11$ .

3 {(x - 2)}^{2} - 11

$3 {(x - 2)}^{2} - 11$

3 {(x - 2)}^{2} - 11

$3 {(x - 2)}^{2} - 11$

Step 1.2

Set

y

$y$ equal to the new right side.

y = 3 {(x - 2)}^{2} - 11

$y = 3 {(x - 2)}^{2} - 11$

y = 3 {(x - 2)}^{2} - 11

$y = 3 {(x - 2)}^{2} - 11$

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 = 3

$a = 3$

h = 2

$h = 2$

k = - 11

$k = - 11$

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(2, - 11)

$(2, - 11)$

Step 4