Pre-Algebra Examples

$f (x) = 2 x^{2} - 16 x + 30$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $2 x^{2} - 16 x + 30$ .

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Step 1.1.1.1

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

$a = 2$

$b = - 16$

$c = 30$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{- 16}{2 \cdot 2}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of $- 16$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $- 16$ .

$d = \frac{2 \cdot - 8}{2 \cdot 2}$

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

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

$d = \frac{2 \cdot - 8}{2 (2)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 8}{2 \cdot 2}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 8}{2}$

Step 1.1.1.3.2.2

Cancel the common factor of $- 8$ and $2$ .

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Step 1.1.1.3.2.2.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2}$

Step 1.1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot - 4}{2 (1)}$

Step 1.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

Step 1.1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 4}{1}$

Step 1.1.1.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = 30 - \frac{{(- 16)}^{2}}{4 \cdot 2}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raise $- 16$ to the power of $2$ .

$e = 30 - \frac{256}{4 \cdot 2}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = 30 - \frac{256}{8}$

Step 1.1.1.4.2.1.3

Divide $256$ by $8$ .

$e = 30 - 1 \cdot 32$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $32$ .

$e = 30 - 32$

Step 1.1.1.4.2.2

Subtract $32$ from $30$ .

$e = - 2$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 4)}^{2} - 2$ .

$2 {(x - 4)}^{2} - 2$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 2 {(x - 4)}^{2} - 2$

Step 1.2

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

$a = 2$

$h = 4$

$k = - 2$

Step 1.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(4, - 2)$

Step 1.5

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

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Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 2}$

Step 1.5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(4, - \frac{15}{8})$

Step 1.7

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

$x = 4$

Step 1.8

Find the directrix.

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Step 1.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.8.2

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

$y = - \frac{17}{8}$

Step 1.9

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

Direction: Opens Up

Vertex: $(4, - 2)$

Focus: $(4, - \frac{15}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{17}{8}$

Direction: Opens Up

Vertex: $(4, - 2)$

Focus: $(4, - \frac{15}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{17}{8}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 2 {(3)}^{2} - 16 \cdot 3 + 30$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raise $3$ to the power of $2$ .

$f (3) = 2 \cdot 9 - 16 \cdot 3 + 30$

Step 2.2.1.2

Multiply $2$ by $9$ .

$f (3) = 18 - 16 \cdot 3 + 30$

Step 2.2.1.3

Multiply $- 16$ by $3$ .

$f (3) = 18 - 48 + 30$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $48$ from $18$ .

$f (3) = - 30 + 30$

Step 2.2.2.2

Add $- 30$ and $30$ .

$f (3) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

The $y$ value at $x = 3$ is $0$ .

$y = 0$

Step 2.4

Replace the variable $x$ with $2$ in the expression.

$f (2) = 2 {(2)}^{2} - 16 \cdot 2 + 30$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 2.5.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 2.5.1.1.1.1

Raise $2$ to the power of $1$ .

$f (2) = 2 {(2)}^{2} - 16 \cdot 2 + 30$

Step 2.5.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = 2^{1 + 2} - 16 \cdot 2 + 30$

Step 2.5.1.1.2

Add $1$ and $2$ .

$f (2) = 2^{3} - 16 \cdot 2 + 30$

Step 2.5.1.2

Raise $2$ to the power of $3$ .

$f (2) = 8 - 16 \cdot 2 + 30$

Step 2.5.1.3

Multiply $- 16$ by $2$ .

$f (2) = 8 - 32 + 30$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $32$ from $8$ .

$f (2) = - 24 + 30$

Step 2.5.2.2

Add $- 24$ and $30$ .

$f (2) = 6$

Step 2.5.3

The final answer is $6$ .

$6$

Step 2.6

The $y$ value at $x = 2$ is $6$ .

$y = 6$

Step 2.7

Replace the variable $x$ with $5$ in the expression.

$f (5) = 2 {(5)}^{2} - 16 \cdot 5 + 30$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Raise $5$ to the power of $2$ .

$f (5) = 2 \cdot 25 - 16 \cdot 5 + 30$

Step 2.8.1.2

Multiply $2$ by $25$ .

$f (5) = 50 - 16 \cdot 5 + 30$

Step 2.8.1.3

Multiply $- 16$ by $5$ .

$f (5) = 50 - 80 + 30$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $80$ from $50$ .

$f (5) = - 30 + 30$

Step 2.8.2.2

Add $- 30$ and $30$ .

$f (5) = 0$

Step 2.8.3

The final answer is $0$ .

$0$

Step 2.9

The $y$ value at $x = 5$ is $0$ .

$y = 0$

Step 2.10

Replace the variable $x$ with $6$ in the expression.

$f (6) = 2 {(6)}^{2} - 16 \cdot 6 + 30$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

Raise $6$ to the power of $2$ .

$f (6) = 2 \cdot 36 - 16 \cdot 6 + 30$

Step 2.11.1.2

Multiply $2$ by $36$ .

$f (6) = 72 - 16 \cdot 6 + 30$

Step 2.11.1.3

Multiply $- 16$ by $6$ .

$f (6) = 72 - 96 + 30$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $96$ from $72$ .

$f (6) = - 24 + 30$

Step 2.11.2.2

Add $- 24$ and $30$ .

$f (6) = 6$

Step 2.11.3

The final answer is $6$ .

$6$

Step 2.12

The $y$ value at $x = 6$ is $6$ .

$y = 6$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 6 \\ 3 & 0 \\ 4 & - 2 \\ 5 & 0 \\ 6 & 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 2)$

Focus: $(4, - \frac{15}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{17}{8}$

$\begin{array}{c} x & y \\ 2 & 6 \\ 3 & 0 \\ 4 & - 2 \\ 5 & 0 \\ 6 & 6 \end{array}$

Step 4