Pre-Algebra Examples

$f (x) = {(x - \frac{3}{2})}^{2} + \frac{5}{2}$

Step 1

Find the properties of the given parabola.

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Step 1.1

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

$a = 1$

$h = \frac{3}{2}$

$k = \frac{5}{2}$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{5}{2})$

Step 1.4

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

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Step 1.4.1

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

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of $a$ into the formula.

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

Step 1.4.3

Cancel the common factor of $1$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.5

Find the focus.

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Step 1.5.1

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 1.5.2

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

$(\frac{3}{2}, \frac{11}{4})$

Step 1.6

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

$x = \frac{3}{2}$

Step 1.7

Find the directrix.

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Step 1.7.1

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 1.7.2

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

$y = \frac{9}{4}$

Step 1.8

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

Direction: Opens Up

Vertex: $(\frac{3}{2}, \frac{5}{2})$

Focus: $(\frac{3}{2}, \frac{11}{4})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = \frac{9}{4}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, \frac{5}{2})$

Focus: $(\frac{3}{2}, \frac{11}{4})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = \frac{9}{4}$

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 $1$ in the expression.

$f (1) = {(1)}^{2} - 3 \cdot 1 + \frac{19}{4}$

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

One to any power is one.

$f (1) = 1 - 3 \cdot 1 + \frac{19}{4}$

Step 2.2.1.2

Multiply $- 3$ by $1$ .

$f (1) = 1 - 3 + \frac{19}{4}$

Step 2.2.2

Find the common denominator.

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Step 2.2.2.1

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

$f (1) = \frac{1}{1} - 3 + \frac{19}{4}$

Step 2.2.2.2

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

$f (1) = \frac{1}{1} \cdot \frac{4}{4} - 3 + \frac{19}{4}$

Step 2.2.2.3

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

$f (1) = \frac{4}{4} - 3 + \frac{19}{4}$

Step 2.2.2.4

Write $- 3$ as a fraction with denominator $1$ .

$f (1) = \frac{4}{4} + \frac{- 3}{1} + \frac{19}{4}$

Step 2.2.2.5

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} + \frac{- 3}{1} \cdot \frac{4}{4} + \frac{19}{4}$

Step 2.2.2.6

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$f (1) = \frac{4}{4} + \frac{- 3 \cdot 4}{4} + \frac{19}{4}$

Step 2.2.3

Combine the numerators over the common denominator.

$f (1) = \frac{4 - 3 \cdot 4 + 19}{4}$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Multiply $- 3$ by $4$ .

$f (1) = \frac{4 - 12 + 19}{4}$

Step 2.2.4.2

Subtract $12$ from $4$ .

$f (1) = \frac{- 8 + 19}{4}$

Step 2.2.4.3

Add $- 8$ and $19$ .

$f (1) = \frac{11}{4}$

Step 2.2.5

The final answer is $\frac{11}{4}$ .

$\frac{11}{4}$

Step 2.3

The $y$ value at $x = 1$ is $\frac{11}{4}$ .

$y = \frac{11}{4}$

Step 2.4

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} - 3 \cdot 0 + \frac{19}{4}$

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

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 3 \cdot 0 + \frac{19}{4}$

Step 2.5.1.2

Multiply $- 3$ by $0$ .

$f (0) = 0 + 0 + \frac{19}{4}$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $0$ and $0$ .

$f (0) = 0 + \frac{19}{4}$

Step 2.5.2.2

Add $0$ and $\frac{19}{4}$ .

$f (0) = \frac{19}{4}$

Step 2.5.3

The final answer is $\frac{19}{4}$ .

$\frac{19}{4}$

Step 2.6

The $y$ value at $x = 0$ is $\frac{19}{4}$ .

$y = \frac{19}{4}$

Step 2.7

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

$f (3) = {(3)}^{2} - 3 \cdot 3 + \frac{19}{4}$

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 $3$ to the power of $2$ .

$f (3) = 9 - 3 \cdot 3 + \frac{19}{4}$

Step 2.8.1.2

Multiply $- 3$ by $3$ .

$f (3) = 9 - 9 + \frac{19}{4}$

Step 2.8.2

Find the common denominator.

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Step 2.8.2.1

Write $9$ as a fraction with denominator $1$ .

$f (3) = \frac{9}{1} - 9 + \frac{19}{4}$

Step 2.8.2.2

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9}{1} \cdot \frac{4}{4} - 9 + \frac{19}{4}$

Step 2.8.2.3

Multiply $\frac{9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} - 9 + \frac{19}{4}$

Step 2.8.2.4

Write $- 9$ as a fraction with denominator $1$ .

$f (3) = \frac{9 \cdot 4}{4} + \frac{- 9}{1} + \frac{19}{4}$

Step 2.8.2.5

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} + \frac{- 9}{1} \cdot \frac{4}{4} + \frac{19}{4}$

Step 2.8.2.6

Multiply $\frac{- 9}{1}$ by $\frac{4}{4}$ .

$f (3) = \frac{9 \cdot 4}{4} + \frac{- 9 \cdot 4}{4} + \frac{19}{4}$

Step 2.8.3

Combine the numerators over the common denominator.

$f (3) = \frac{9 \cdot 4 - 9 \cdot 4 + 19}{4}$

Step 2.8.4

Simplify each term.

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Step 2.8.4.1

Multiply $9$ by $4$ .

$f (3) = \frac{36 - 9 \cdot 4 + 19}{4}$

Step 2.8.4.2

Multiply $- 9$ by $4$ .

$f (3) = \frac{36 - 36 + 19}{4}$

Step 2.8.5

Simplify by adding and subtracting.

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Step 2.8.5.1

Subtract $36$ from $36$ .

$f (3) = \frac{0 + 19}{4}$

Step 2.8.5.2

Add $0$ and $19$ .

$f (3) = \frac{19}{4}$

Step 2.8.6

The final answer is $\frac{19}{4}$ .

$\frac{19}{4}$

Step 2.9

The $y$ value at $x = 3$ is $\frac{19}{4}$ .

$y = \frac{19}{4}$

Step 2.10

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 3 \cdot 4 + \frac{19}{4}$

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 $4$ to the power of $2$ .

$f (4) = 16 - 3 \cdot 4 + \frac{19}{4}$

Step 2.11.1.2

Multiply $- 3$ by $4$ .

$f (4) = 16 - 12 + \frac{19}{4}$

Step 2.11.2

Find the common denominator.

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Step 2.11.2.1

Write $16$ as a fraction with denominator $1$ .

$f (4) = \frac{16}{1} - 12 + \frac{19}{4}$

Step 2.11.2.2

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$f (4) = \frac{16}{1} \cdot \frac{4}{4} - 12 + \frac{19}{4}$

Step 2.11.2.3

Multiply $\frac{16}{1}$ by $\frac{4}{4}$ .

$f (4) = \frac{16 \cdot 4}{4} - 12 + \frac{19}{4}$

Step 2.11.2.4

Write $- 12$ as a fraction with denominator $1$ .

$f (4) = \frac{16 \cdot 4}{4} + \frac{- 12}{1} + \frac{19}{4}$

Step 2.11.2.5

Multiply $\frac{- 12}{1}$ by $\frac{4}{4}$ .

$f (4) = \frac{16 \cdot 4}{4} + \frac{- 12}{1} \cdot \frac{4}{4} + \frac{19}{4}$

Step 2.11.2.6

Multiply $\frac{- 12}{1}$ by $\frac{4}{4}$ .

$f (4) = \frac{16 \cdot 4}{4} + \frac{- 12 \cdot 4}{4} + \frac{19}{4}$

Step 2.11.3

Combine the numerators over the common denominator.

$f (4) = \frac{16 \cdot 4 - 12 \cdot 4 + 19}{4}$

Step 2.11.4

Simplify each term.

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Step 2.11.4.1

Multiply $16$ by $4$ .

$f (4) = \frac{64 - 12 \cdot 4 + 19}{4}$

Step 2.11.4.2

Multiply $- 12$ by $4$ .

$f (4) = \frac{64 - 48 + 19}{4}$

Step 2.11.5

Simplify by adding and subtracting.

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Step 2.11.5.1

Subtract $48$ from $64$ .

$f (4) = \frac{16 + 19}{4}$

Step 2.11.5.2

Add $16$ and $19$ .

$f (4) = \frac{35}{4}$

Step 2.11.6

The final answer is $\frac{35}{4}$ .

$\frac{35}{4}$

Step 2.12

The $y$ value at $x = 4$ is $\frac{35}{4}$ .

$y = \frac{35}{4}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & \frac{19}{4} \\ 1 & \frac{11}{4} \\ \frac{3}{2} & \frac{5}{2} \\ 3 & \frac{19}{4} \\ 4 & \frac{35}{4} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, \frac{5}{2})$

Focus: $(\frac{3}{2}, \frac{11}{4})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = \frac{9}{4}$

$\begin{array}{c} x & y \\ 0 & \frac{19}{4} \\ 1 & \frac{11}{4} \\ \frac{3}{2} & \frac{5}{2} \\ 3 & \frac{19}{4} \\ 4 & \frac{35}{4} \end{array}$

Step 4