Pre-Algebra Examples

$t^{2} + \frac{2}{3} t + \frac{1}{9}$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$y = x^{2} + \frac{2 x}{3} + \frac{1}{9}$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $x^{2} + \frac{2 x}{3} + \frac{1}{9}$ .

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Step 2.1.1.1

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

$a = 1$

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

$c = \frac{1}{9}$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.2

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 2.1.1.3.2.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{1}{3}$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = \frac{1}{9} - \frac{{(\frac{2}{3})}^{2}}{4 \cdot 1}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Simplify the numerator.

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Step 2.1.1.4.2.1.1.1

Apply the product rule to $\frac{2}{3}$ .

$e = \frac{1}{9} - \frac{\frac{2^{2}}{3^{2}}}{4 \cdot 1}$

Step 2.1.1.4.2.1.1.2

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

$e = \frac{1}{9} - \frac{\frac{4}{3^{2}}}{4 \cdot 1}$

Step 2.1.1.4.2.1.1.3

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

$e = \frac{1}{9} - \frac{\frac{4}{9}}{4 \cdot 1}$

Step 2.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = \frac{1}{9} - \frac{\frac{4}{9}}{4}$

Step 2.1.1.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{9} - (\frac{4}{9} \cdot \frac{1}{4})$

Step 2.1.1.4.2.1.4

Cancel the common factor of $4$ .

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Step 2.1.1.4.2.1.4.1

Cancel the common factor.

$e = \frac{1}{9} - (\frac{4}{9} \cdot \frac{1}{4})$

Step 2.1.1.4.2.1.4.2

Rewrite the expression.

$e = \frac{1}{9} - \frac{1}{9}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 - 1}{9}$

Step 2.1.1.4.2.3

Subtract $1$ from $1$ .

$e = \frac{0}{9}$

Step 2.1.1.4.2.4

Divide $0$ by $9$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{3})}^{2} + 0$ .

${(x + \frac{1}{3})}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = {(x + \frac{1}{3})}^{2} + 0$

Step 2.2

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

$a = 1$

$h = - \frac{1}{3}$

$k = 0$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(- \frac{1}{3}, 0)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Cancel the common factor of $1$ .

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Step 2.5.3.1

Cancel the common factor.

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

Step 2.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(- \frac{1}{3}, \frac{1}{4})$

Step 2.7

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

$x = - \frac{1}{3}$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

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

Step 2.9

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

Direction: Opens Up

Vertex: $(- \frac{1}{3}, 0)$

Focus: $(- \frac{1}{3}, \frac{1}{4})$

Axis of Symmetry: $x = - \frac{1}{3}$

Directrix: $y = - \frac{1}{4}$

Direction: Opens Up

Vertex: $(- \frac{1}{3}, 0)$

Focus: $(- \frac{1}{3}, \frac{1}{4})$

Axis of Symmetry: $x = - \frac{1}{3}$

Directrix: $y = - \frac{1}{4}$

Step 3

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 3.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = {(- 1)}^{2} + \frac{2 (- 1)}{3} + \frac{1}{9}$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $- 1$ .

$f (- 1) = {(- 1)}^{2} + \frac{- 2}{3} + \frac{1}{9}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (- 1) = 1 + \frac{- 2}{3} + \frac{1}{9}$

Step 3.2.2.2

Move the negative in front of the fraction.

$f (- 1) = 1 - \frac{2}{3} + \frac{1}{9}$

Step 3.2.3

Find the common denominator.

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Step 3.2.3.1

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

$f (- 1) = \frac{1}{1} - \frac{2}{3} + \frac{1}{9}$

Step 3.2.3.2

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

$f (- 1) = \frac{1}{1} \cdot \frac{9}{9} - \frac{2}{3} + \frac{1}{9}$

Step 3.2.3.3

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

$f (- 1) = \frac{9}{9} - \frac{2}{3} + \frac{1}{9}$

Step 3.2.3.4

Multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$f (- 1) = \frac{9}{9} - (\frac{2}{3} \cdot \frac{3}{3}) + \frac{1}{9}$

Step 3.2.3.5

Multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$f (- 1) = \frac{9}{9} - \frac{2 \cdot 3}{3 \cdot 3} + \frac{1}{9}$

Step 3.2.3.6

Multiply $3$ by $3$ .

$f (- 1) = \frac{9}{9} - \frac{2 \cdot 3}{9} + \frac{1}{9}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{9 - 2 \cdot 3 + 1}{9}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Multiply $- 2$ by $3$ .

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

Step 3.2.5.2

Subtract $6$ from $9$ .

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

Step 3.2.5.3

Add $3$ and $1$ .

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

Step 3.2.6

The final answer is $\frac{4}{9}$ .

$\frac{4}{9}$

Step 3.3

The $y$ value at $x = - 1$ is $\frac{4}{9}$ .

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

Step 3.4

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

$f (- 2) = {(- 2)}^{2} + \frac{2 (- 2)}{3} + \frac{1}{9}$

Step 3.5

Simplify the result.

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Step 3.5.1

Multiply $2$ by $- 2$ .

$f (- 2) = {(- 2)}^{2} + \frac{- 4}{3} + \frac{1}{9}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (- 2) = 4 + \frac{- 4}{3} + \frac{1}{9}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f (- 2) = 4 - \frac{4}{3} + \frac{1}{9}$

Step 3.5.3

Find the common denominator.

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Step 3.5.3.1

Write $4$ as a fraction with denominator $1$ .

$f (- 2) = \frac{4}{1} - \frac{4}{3} + \frac{1}{9}$

Step 3.5.3.2

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

$f (- 2) = \frac{4}{1} \cdot \frac{9}{9} - \frac{4}{3} + \frac{1}{9}$

Step 3.5.3.3

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

$f (- 2) = \frac{4 \cdot 9}{9} - \frac{4}{3} + \frac{1}{9}$

Step 3.5.3.4

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

$f (- 2) = \frac{4 \cdot 9}{9} - (\frac{4}{3} \cdot \frac{3}{3}) + \frac{1}{9}$

Step 3.5.3.5

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

$f (- 2) = \frac{4 \cdot 9}{9} - \frac{4 \cdot 3}{3 \cdot 3} + \frac{1}{9}$

Step 3.5.3.6

Multiply $3$ by $3$ .

$f (- 2) = \frac{4 \cdot 9}{9} - \frac{4 \cdot 3}{9} + \frac{1}{9}$

Step 3.5.4

Combine the numerators over the common denominator.

$f (- 2) = \frac{4 \cdot 9 - 4 \cdot 3 + 1}{9}$

Step 3.5.5

Simplify each term.

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Step 3.5.5.1

Multiply $4$ by $9$ .

$f (- 2) = \frac{36 - 4 \cdot 3 + 1}{9}$

Step 3.5.5.2

Multiply $- 4$ by $3$ .

$f (- 2) = \frac{36 - 12 + 1}{9}$

Step 3.5.6

Simplify by adding and subtracting.

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Step 3.5.6.1

Subtract $12$ from $36$ .

$f (- 2) = \frac{24 + 1}{9}$

Step 3.5.6.2

Add $24$ and $1$ .

$f (- 2) = \frac{25}{9}$

Step 3.5.7

The final answer is $\frac{25}{9}$ .

$\frac{25}{9}$

Step 3.6

The $y$ value at $x = - 2$ is $\frac{25}{9}$ .

$y = \frac{25}{9}$

Step 3.7

Replace the variable $x$ with $1$ in the expression.

$f (1) = {(1)}^{2} + \frac{2 (1)}{3} + \frac{1}{9}$

Step 3.8

Simplify the result.

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Step 3.8.1

Multiply $2$ by $1$ .

$f (1) = {(1)}^{2} + \frac{2}{3} + \frac{1}{9}$

Step 3.8.2

One to any power is one.

$f (1) = 1 + \frac{2}{3} + \frac{1}{9}$

Step 3.8.3

Find the common denominator.

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Step 3.8.3.1

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

$f (1) = \frac{1}{1} + \frac{2}{3} + \frac{1}{9}$

Step 3.8.3.2

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

$f (1) = \frac{1}{1} \cdot \frac{9}{9} + \frac{2}{3} + \frac{1}{9}$

Step 3.8.3.3

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

$f (1) = \frac{9}{9} + \frac{2}{3} + \frac{1}{9}$

Step 3.8.3.4

Multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$f (1) = \frac{9}{9} + \frac{2}{3} \cdot \frac{3}{3} + \frac{1}{9}$

Step 3.8.3.5

Multiply $\frac{2}{3}$ by $\frac{3}{3}$ .

$f (1) = \frac{9}{9} + \frac{2 \cdot 3}{3 \cdot 3} + \frac{1}{9}$

Step 3.8.3.6

Multiply $3$ by $3$ .

$f (1) = \frac{9}{9} + \frac{2 \cdot 3}{9} + \frac{1}{9}$

Step 3.8.4

Combine the numerators over the common denominator.

$f (1) = \frac{9 + 2 \cdot 3 + 1}{9}$

Step 3.8.5

Simplify the expression.

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Step 3.8.5.1

Multiply $2$ by $3$ .

$f (1) = \frac{9 + 6 + 1}{9}$

Step 3.8.5.2

Add $9$ and $6$ .

$f (1) = \frac{15 + 1}{9}$

Step 3.8.5.3

Add $15$ and $1$ .

$f (1) = \frac{16}{9}$

Step 3.8.6

The final answer is $\frac{16}{9}$ .

$\frac{16}{9}$

Step 3.9

The $y$ value at $x = 1$ is $\frac{16}{9}$ .

$y = \frac{16}{9}$

Step 3.10

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

$f (2) = {(2)}^{2} + \frac{2 (2)}{3} + \frac{1}{9}$

Step 3.11

Simplify the result.

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Step 3.11.1

Multiply $2$ by $2$ .

$f (2) = {(2)}^{2} + \frac{4}{3} + \frac{1}{9}$

Step 3.11.2

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

$f (2) = 4 + \frac{4}{3} + \frac{1}{9}$

Step 3.11.3

Find the common denominator.

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Step 3.11.3.1

Write $4$ as a fraction with denominator $1$ .

$f (2) = \frac{4}{1} + \frac{4}{3} + \frac{1}{9}$

Step 3.11.3.2

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

$f (2) = \frac{4}{1} \cdot \frac{9}{9} + \frac{4}{3} + \frac{1}{9}$

Step 3.11.3.3

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

$f (2) = \frac{4 \cdot 9}{9} + \frac{4}{3} + \frac{1}{9}$

Step 3.11.3.4

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

$f (2) = \frac{4 \cdot 9}{9} + \frac{4}{3} \cdot \frac{3}{3} + \frac{1}{9}$

Step 3.11.3.5

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

$f (2) = \frac{4 \cdot 9}{9} + \frac{4 \cdot 3}{3 \cdot 3} + \frac{1}{9}$

Step 3.11.3.6

Multiply $3$ by $3$ .

$f (2) = \frac{4 \cdot 9}{9} + \frac{4 \cdot 3}{9} + \frac{1}{9}$

Step 3.11.4

Combine the numerators over the common denominator.

$f (2) = \frac{4 \cdot 9 + 4 \cdot 3 + 1}{9}$

Step 3.11.5

Simplify each term.

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Step 3.11.5.1

Multiply $4$ by $9$ .

$f (2) = \frac{36 + 4 \cdot 3 + 1}{9}$

Step 3.11.5.2

Multiply $4$ by $3$ .

$f (2) = \frac{36 + 12 + 1}{9}$

Step 3.11.6

Simplify by adding numbers.

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Step 3.11.6.1

Add $36$ and $12$ .

$f (2) = \frac{48 + 1}{9}$

Step 3.11.6.2

Add $48$ and $1$ .

$f (2) = \frac{49}{9}$

Step 3.11.7

The final answer is $\frac{49}{9}$ .

$\frac{49}{9}$

Step 3.12

The $y$ value at $x = 2$ is $\frac{49}{9}$ .

$y = \frac{49}{9}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & \frac{25}{9} \\ - 1 & \frac{4}{9} \\ - \frac{1}{3} & 0 \\ 1 & \frac{16}{9} \\ 2 & \frac{49}{9} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- \frac{1}{3}, 0)$

Focus: $(- \frac{1}{3}, \frac{1}{4})$

Axis of Symmetry: $x = - \frac{1}{3}$

Directrix: $y = - \frac{1}{4}$

$\begin{array}{c} x & y \\ - 2 & \frac{25}{9} \\ - 1 & \frac{4}{9} \\ - \frac{1}{3} & 0 \\ 1 & \frac{16}{9} \\ 2 & \frac{49}{9} \end{array}$

Step 5