Algebra Examples

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

Step 1

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

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

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $- 3 x^{2} + 12 x - 6$ .

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Step 2.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 3$

$b = 12$

$c = - 6$

Step 2.1.2

Consider the vertex form of a parabola.

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

Step 2.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Cancel the common factor of $12$ and $2$ .

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Step 2.1.3.2.1.1

Factor $2$ out of $12$ .

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

Step 2.1.3.2.1.2

Cancel the common factors.

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Step 2.1.3.2.1.2.1

Factor $2$ out of $2 \cdot - 3$ .

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

Step 2.1.3.2.1.2.2

Cancel the common factor.

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

Step 2.1.3.2.1.2.3

Rewrite the expression.

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

Step 2.1.3.2.2

Cancel the common factor of $6$ and $- 3$ .

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Step 2.1.3.2.2.1

Factor $3$ out of $6$ .

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

Step 2.1.3.2.2.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 2.1.3.2.3

Multiply $- 1$ by $2$ .

$d = - 2$

Step 2.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

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 $12$ to the power of $2$ .

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

Step 2.1.4.2.1.2

Multiply $4$ by $- 3$ .

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

Step 2.1.4.2.1.3

Divide $144$ by $- 12$ .

$e = - 6 - - 12$

Step 2.1.4.2.1.4

Multiply $- 1$ by $- 12$ .

$e = - 6 + 12$

Step 2.1.4.2.2

Add $- 6$ and $12$ .

$e = 6$

Step 2.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - 2)}^{2} + 6$ .

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

Step 2.2

Set $y$ equal to the new right side.

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

Step 3

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

$a = - 3$

$h = 2$

$k = 6$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(2, 6)$

Step 6

Find $p$ , 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.

$\frac{1}{4 a}$

Step 6.2

Substitute the value of $a$ into the formula.

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Step 6.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 7.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

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

Step 8

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

$x = 2$

Step 9