Pre-Algebra Examples

$f (x) = \frac{1}{4} \cdot {(x + 4)}^{2} + 3$

Step 1

Find the properties of the given parabola.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Combine $\frac{1}{4}$ and ${(x + 4)}^{2}$ .

$y = \frac{{(x + 4)}^{2}}{4} + 3$

Step 1.1.2

Reorder terms.

$y = \frac{1}{4} \cdot {(x + 4)}^{2} + 3$

Step 1.2

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

$a = \frac{1}{4}$

$h = - 4$

$k = 3$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 4, 3)$

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.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Combine $4$ and $\frac{1}{4}$ .

$\frac{1}{\frac{4}{4}}$

Step 1.5.3.2

Simplify by dividing numbers.

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Step 1.5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 1.5.3.2.2

Divide $1$ by $1$ .

$1$

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.

$(- 4, 4)$

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.

$y = 2$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 4, 3)$

Focus: $(- 4, 4)$

Axis of Symmetry: $x = - 4$

Directrix: $y = 2$

Direction: Opens Up

Vertex: $(- 4, 3)$

Focus: $(- 4, 4)$

Axis of Symmetry: $x = - 4$

Directrix: $y = 2$

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 $- 6$ in the expression.

$f (- 6) = \frac{{(- 6)}^{2}}{4} + 2 (- 6) + 7$

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

$f (- 6) = \frac{36}{4} + 2 (- 6) + 7$

Step 2.2.1.2

Divide $36$ by $4$ .

$f (- 6) = 9 + 2 (- 6) + 7$

Step 2.2.1.3

Multiply $2$ by $- 6$ .

$f (- 6) = 9 - 12 + 7$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $12$ from $9$ .

$f (- 6) = - 3 + 7$

Step 2.2.2.2

Add $- 3$ and $7$ .

$f (- 6) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 2.3

The $y$ value at $x = - 6$ is $4$ .

$y = 4$

Step 2.4

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

$f (- 5) = \frac{{(- 5)}^{2}}{4} + 2 (- 5) + 7$

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

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

$f (- 5) = \frac{25}{4} + 2 (- 5) + 7$

Step 2.5.1.2

Multiply $2$ by $- 5$ .

$f (- 5) = \frac{25}{4} - 10 + 7$

Step 2.5.2

Find the common denominator.

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Step 2.5.2.1

Write $- 10$ as a fraction with denominator $1$ .

$f (- 5) = \frac{25}{4} + \frac{- 10}{1} + 7$

Step 2.5.2.2

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10}{1} \cdot \frac{4}{4} + 7$

Step 2.5.2.3

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + 7$

Step 2.5.2.4

Write $7$ as a fraction with denominator $1$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{7}{1}$

Step 2.5.2.5

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{7}{1} \cdot \frac{4}{4}$

Step 2.5.2.6

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{7 \cdot 4}{4}$

Step 2.5.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{25 - 10 \cdot 4 + 7 \cdot 4}{4}$

Step 2.5.4

Simplify each term.

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Step 2.5.4.1

Multiply $- 10$ by $4$ .

$f (- 5) = \frac{25 - 40 + 7 \cdot 4}{4}$

Step 2.5.4.2

Multiply $7$ by $4$ .

$f (- 5) = \frac{25 - 40 + 28}{4}$

Step 2.5.5

Simplify by adding and subtracting.

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Step 2.5.5.1

Subtract $40$ from $25$ .

$f (- 5) = \frac{- 15 + 28}{4}$

Step 2.5.5.2

Add $- 15$ and $28$ .

$f (- 5) = \frac{13}{4}$

Step 2.5.6

The final answer is $\frac{13}{4}$ .

$\frac{13}{4}$

Step 2.6

The $y$ value at $x = - 5$ is $\frac{13}{4}$ .

$y = \frac{13}{4}$

Step 2.7

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

$f (- 2) = \frac{{(- 2)}^{2}}{4} + 2 (- 2) + 7$

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

$f (- 2) = \frac{4}{4} + 2 (- 2) + 7$

Step 2.8.1.2

Divide $4$ by $4$ .

$f (- 2) = 1 + 2 (- 2) + 7$

Step 2.8.1.3

Multiply $2$ by $- 2$ .

$f (- 2) = 1 - 4 + 7$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $4$ from $1$ .

$f (- 2) = - 3 + 7$

Step 2.8.2.2

Add $- 3$ and $7$ .

$f (- 2) = 4$

Step 2.8.3

The final answer is $4$ .

$4$

Step 2.9

The $y$ value at $x = - 2$ is $4$ .

$y = 4$

Step 2.10

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

$f (- 3) = \frac{{(- 3)}^{2}}{4} + 2 (- 3) + 7$

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

$f (- 3) = \frac{9}{4} + 2 (- 3) + 7$

Step 2.11.1.2

Multiply $2$ by $- 3$ .

$f (- 3) = \frac{9}{4} - 6 + 7$

Step 2.11.2

Find the common denominator.

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Step 2.11.2.1

Write $- 6$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{4} + \frac{- 6}{1} + 7$

Step 2.11.2.2

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6}{1} \cdot \frac{4}{4} + 7$

Step 2.11.2.3

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + 7$

Step 2.11.2.4

Write $7$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{7}{1}$

Step 2.11.2.5

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{7}{1} \cdot \frac{4}{4}$

Step 2.11.2.6

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{7 \cdot 4}{4}$

Step 2.11.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 - 6 \cdot 4 + 7 \cdot 4}{4}$

Step 2.11.4

Simplify each term.

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Step 2.11.4.1

Multiply $- 6$ by $4$ .

$f (- 3) = \frac{9 - 24 + 7 \cdot 4}{4}$

Step 2.11.4.2

Multiply $7$ by $4$ .

$f (- 3) = \frac{9 - 24 + 28}{4}$

Step 2.11.5

Simplify by adding and subtracting.

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Step 2.11.5.1

Subtract $24$ from $9$ .

$f (- 3) = \frac{- 15 + 28}{4}$

Step 2.11.5.2

Add $- 15$ and $28$ .

$f (- 3) = \frac{13}{4}$

Step 2.11.6

The final answer is $\frac{13}{4}$ .

$\frac{13}{4}$

Step 2.12

The $y$ value at $x = - 3$ is $\frac{13}{4}$ .

$y = \frac{13}{4}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & 4 \\ - 5 & \frac{13}{4} \\ - 4 & 3 \\ - 3 & \frac{13}{4} \\ - 2 & 4 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 4, 3)$

Focus: $(- 4, 4)$

Axis of Symmetry: $x = - 4$

Directrix: $y = 2$

$\begin{array}{c} x & y \\ - 6 & 4 \\ - 5 & \frac{13}{4} \\ - 4 & 3 \\ - 3 & \frac{13}{4} \\ - 2 & 4 \end{array}$

Step 4