Pre-Algebra Examples

$x^{2} - 4 x - 6 y + 13 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$- 4 x - 6 y + 13 = - x^{2}$

Step 1.1.2

Add $4 x$ to both sides of the equation.

$- 6 y + 13 = - x^{2} + 4 x$

Step 1.1.3

Subtract $13$ from both sides of the equation.

$- 6 y = - x^{2} + 4 x - 13$

Step 1.2

Divide each term in $- 6 y = - x^{2} + 4 x - 13$ by $- 6$ and simplify.

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Step 1.2.1

Divide each term in $- 6 y = - x^{2} + 4 x - 13$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- x^{2}}{- 6} + \frac{4 x}{- 6} + \frac{- 13}{- 6}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- x^{2}}{- 6} + \frac{4 x}{- 6} + \frac{- 13}{- 6}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 6} + \frac{4 x}{- 6} + \frac{- 13}{- 6}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{6} + \frac{4 x}{- 6} + \frac{- 13}{- 6}$

Step 1.2.3.1.2

Cancel the common factor of $4$ and $- 6$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $4 x$ .

$y = \frac{x^{2}}{6} + \frac{2 (2 x)}{- 6} + \frac{- 13}{- 6}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $- 6$ .

$y = \frac{x^{2}}{6} + \frac{2 (2 x)}{2 (- 3)} + \frac{- 13}{- 6}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{6} + \frac{2 (2 x)}{2 \cdot - 3} + \frac{- 13}{- 6}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{6} + \frac{2 x}{- 3} + \frac{- 13}{- 6}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{- 13}{- 6}$

Step 1.2.3.1.4

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{6} - \frac{2 x}{3} + \frac{13}{6}$

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 $\frac{x^{2}}{6} - \frac{2 x}{3} + \frac{13}{6}$ .

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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 $c$ .

$a = \frac{1}{6}$

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

$c = \frac{13}{6}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{2}{3}}{2 (\frac{1}{6})}$

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.

$d = - \frac{2}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

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

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$d = \frac{- 2}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

Step 2.1.1.3.2.2.2

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

Step 2.1.1.3.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot - 1}{3} \cdot \frac{1}{2 (\frac{1}{6})}$

Step 2.1.1.3.2.2.4

Rewrite the expression.

$d = \frac{- 1}{3} \cdot \frac{1}{\frac{1}{6}}$

Step 2.1.1.3.2.3

Multiply $\frac{- 1}{3}$ by $\frac{1}{\frac{1}{6}}$ .

$d = \frac{- 1}{3 (\frac{1}{6})}$

Step 2.1.1.3.2.4

Combine $3$ and $\frac{1}{6}$ .

$d = \frac{- 1}{\frac{3}{6}}$

Step 2.1.1.3.2.5

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

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Step 2.1.1.3.2.5.1

Factor $3$ out of $3$ .

$d = \frac{- 1}{\frac{3 (1)}{6}}$

Step 2.1.1.3.2.5.2

Cancel the common factors.

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Step 2.1.1.3.2.5.2.1

Factor $3$ out of $6$ .

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

Step 2.1.1.3.2.5.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.5.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Step 2.1.1.3.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{13}{6} - \frac{{(- \frac{2}{3})}^{2}}{4 (\frac{1}{6})}$

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 $- \frac{2}{3}$ .

$e = \frac{13}{6} - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.1.2

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

$e = \frac{13}{6} - \frac{1 {(\frac{2}{3})}^{2}}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.1.3

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

$e = \frac{13}{6} - \frac{1 \frac{2^{2}}{3^{2}}}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.1.4

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

$e = \frac{13}{6} - \frac{1 \frac{4}{3^{2}}}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.1.5

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

$e = \frac{13}{6} - \frac{1 (\frac{4}{9})}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.1.6

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

$e = \frac{13}{6} - \frac{\frac{4}{9}}{4 (\frac{1}{6})}$

Step 2.1.1.4.2.1.2

Combine $4$ and $\frac{1}{6}$ .

$e = \frac{13}{6} - \frac{\frac{4}{9}}{\frac{4}{6}}$

Step 2.1.1.4.2.1.3

Cancel the common factor of $4$ and $6$ .

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Step 2.1.1.4.2.1.3.1

Factor $2$ out of $4$ .

$e = \frac{13}{6} - \frac{\frac{4}{9}}{\frac{2 (2)}{6}}$

Step 2.1.1.4.2.1.3.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.2.1

Factor $2$ out of $6$ .

$e = \frac{13}{6} - \frac{\frac{4}{9}}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 2.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = \frac{13}{6} - \frac{\frac{4}{9}}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 2.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = \frac{13}{6} - \frac{\frac{4}{9}}{\frac{2}{3}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{13}{6} - (\frac{4}{9} \cdot \frac{3}{2})$

Step 2.1.1.4.2.1.5

Cancel the common factor of $2$ .

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Step 2.1.1.4.2.1.5.1

Factor $2$ out of $4$ .

$e = \frac{13}{6} - (\frac{2 (2)}{9} \cdot \frac{3}{2})$

Step 2.1.1.4.2.1.5.2

Cancel the common factor.

$e = \frac{13}{6} - (\frac{2 \cdot 2}{9} \cdot \frac{3}{2})$

Step 2.1.1.4.2.1.5.3

Rewrite the expression.

$e = \frac{13}{6} - (\frac{2}{9} \cdot 3)$

Step 2.1.1.4.2.1.6

Cancel the common factor of $3$ .

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Step 2.1.1.4.2.1.6.1

Factor $3$ out of $9$ .

$e = \frac{13}{6} - (\frac{2}{3 (3)} \cdot 3)$

Step 2.1.1.4.2.1.6.2

Cancel the common factor.

$e = \frac{13}{6} - (\frac{2}{3 \cdot 3} \cdot 3)$

Step 2.1.1.4.2.1.6.3

Rewrite the expression.

$e = \frac{13}{6} - \frac{2}{3}$

Step 2.1.1.4.2.2

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = \frac{13}{6} - \frac{2}{3} \cdot \frac{2}{2}$

Step 2.1.1.4.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.1.4.2.3.1

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

$e = \frac{13}{6} - \frac{2 \cdot 2}{3 \cdot 2}$

Step 2.1.1.4.2.3.2

Multiply $3$ by $2$ .

$e = \frac{13}{6} - \frac{2 \cdot 2}{6}$

Step 2.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{13 - 2 \cdot 2}{6}$

Step 2.1.1.4.2.5

Simplify the numerator.

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Step 2.1.1.4.2.5.1

Multiply $- 2$ by $2$ .

$e = \frac{13 - 4}{6}$

Step 2.1.1.4.2.5.2

Subtract $4$ from $13$ .

$e = \frac{9}{6}$

Step 2.1.1.4.2.6

Cancel the common factor of $9$ and $6$ .

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Step 2.1.1.4.2.6.1

Factor $3$ out of $9$ .

$e = \frac{3 (3)}{6}$

Step 2.1.1.4.2.6.2

Cancel the common factors.

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Step 2.1.1.4.2.6.2.1

Factor $3$ out of $6$ .

$e = \frac{3 \cdot 3}{3 \cdot 2}$

Step 2.1.1.4.2.6.2.2

Cancel the common factor.

$e = \frac{3 \cdot 3}{3 \cdot 2}$

Step 2.1.1.4.2.6.2.3

Rewrite the expression.

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

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{6} {(x - 2)}^{2} + \frac{3}{2}$ .

$\frac{1}{6} {(x - 2)}^{2} + \frac{3}{2}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{6} \cdot {(x - 2)}^{2} + \frac{3}{2}$

Step 2.2

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

$a = \frac{1}{6}$

$h = 2$

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

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

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

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.

$\frac{1}{4 \cdot \frac{1}{6}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{1}{6}$ .

$\frac{1}{\frac{4}{6}}$

Step 2.5.3.2

Cancel the common factor of $4$ and $6$ .

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Step 2.5.3.2.1

Factor $2$ out of $4$ .

$\frac{1}{\frac{2 (2)}{6}}$

Step 2.5.3.2.2

Cancel the common factors.

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Step 2.5.3.2.2.1

Factor $2$ out of $6$ .

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

Step 2.5.3.2.2.2

Cancel the common factor.

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

Step 2.5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{2}{3}}$

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{2})$

Step 2.5.3.4

Multiply $\frac{3}{2}$ by $1$ .

$\frac{3}{2}$

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.

$(2, 3)$

Step 2.7

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

$x = 2$

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.

$y = 0$

Step 2.9

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

Direction: Opens Up

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

Focus: $(2, 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = 0$

Direction: Opens Up

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

Focus: $(2, 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = 0$

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.

$f (1) = \frac{{(1)}^{2}}{6} - \frac{2 (1)}{3} + \frac{13}{6}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine fractions.

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Step 3.2.1.1

Combine the numerators over the common denominator.

$f (1) = \frac{{(1)}^{2} + 13}{6} + \frac{- 2 \cdot 1}{3}$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

One to any power is one.

$f (1) = \frac{1 + 13}{6} + \frac{- 2 \cdot 1}{3}$

Step 3.2.1.2.2

Add $1$ and $13$ .

$f (1) = \frac{14}{6} + \frac{- 2 \cdot 1}{3}$

Step 3.2.1.2.3

Multiply $- 2$ by $1$ .

$f (1) = \frac{14}{6} + \frac{- 2}{3}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Cancel the common factor of $14$ and $6$ .

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Step 3.2.2.1.1

Factor $2$ out of $14$ .

$f (1) = \frac{2 (7)}{6} + \frac{- 2}{3}$

Step 3.2.2.1.2

Cancel the common factors.

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Step 3.2.2.1.2.1

Factor $2$ out of $6$ .

$f (1) = \frac{2 \cdot 7}{2 \cdot 3} + \frac{- 2}{3}$

Step 3.2.2.1.2.2

Cancel the common factor.

$f (1) = \frac{2 \cdot 7}{2 \cdot 3} + \frac{- 2}{3}$

Step 3.2.2.1.2.3

Rewrite the expression.

$f (1) = \frac{7}{3} + \frac{- 2}{3}$

Step 3.2.2.2

Move the negative in front of the fraction.

$f (1) = \frac{7}{3} - \frac{2}{3}$

Step 3.2.3

Combine fractions.

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Step 3.2.3.1

Combine the numerators over the common denominator.

$f (1) = \frac{7 - 2}{3}$

Step 3.2.3.2

Subtract $2$ from $7$ .

$f (1) = \frac{5}{3}$

Step 3.2.4

The final answer is $\frac{5}{3}$ .

$\frac{5}{3}$

Step 3.3

The $y$ value at $x = 1$ is $\frac{5}{3}$ .

$y = \frac{5}{3}$

Step 3.4

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

$f (0) = \frac{{(0)}^{2}}{6} - \frac{2 (0)}{3} + \frac{13}{6}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (0) = \frac{{(0)}^{2} + 13}{6} + \frac{- 2 \cdot 0}{3}$

Step 3.5.2

Simplify the expression.

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Step 3.5.2.1

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

$f (0) = \frac{0 + 13}{6} + \frac{- 2 \cdot 0}{3}$

Step 3.5.2.2

Add $0$ and $13$ .

$f (0) = \frac{13}{6} + \frac{- 2 \cdot 0}{3}$

Step 3.5.2.3

Multiply $- 2$ by $0$ .

$f (0) = \frac{13}{6} + \frac{0}{3}$

Step 3.5.2.4

Divide $0$ by $3$ .

$f (0) = \frac{13}{6} + 0$

Step 3.5.2.5

Add $\frac{13}{6}$ and $0$ .

$f (0) = \frac{13}{6}$

Step 3.5.3

The final answer is $\frac{13}{6}$ .

$\frac{13}{6}$

Step 3.6

The $y$ value at $x = 0$ is $\frac{13}{6}$ .

$y = \frac{13}{6}$

Step 3.7

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

$f (5) = \frac{{(5)}^{2}}{6} - \frac{2 (5)}{3} + \frac{13}{6}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine fractions.

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Step 3.8.1.1

Combine the numerators over the common denominator.

$f (5) = \frac{{(5)}^{2} + 13}{6} + \frac{- 2 \cdot 5}{3}$

Step 3.8.1.2

Simplify the expression.

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Step 3.8.1.2.1

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

$f (5) = \frac{25 + 13}{6} + \frac{- 2 \cdot 5}{3}$

Step 3.8.1.2.2

Add $25$ and $13$ .

$f (5) = \frac{38}{6} + \frac{- 2 \cdot 5}{3}$

Step 3.8.1.2.3

Multiply $- 2$ by $5$ .

$f (5) = \frac{38}{6} + \frac{- 10}{3}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Cancel the common factor of $38$ and $6$ .

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Step 3.8.2.1.1

Factor $2$ out of $38$ .

$f (5) = \frac{2 (19)}{6} + \frac{- 10}{3}$

Step 3.8.2.1.2

Cancel the common factors.

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Step 3.8.2.1.2.1

Factor $2$ out of $6$ .

$f (5) = \frac{2 \cdot 19}{2 \cdot 3} + \frac{- 10}{3}$

Step 3.8.2.1.2.2

Cancel the common factor.

$f (5) = \frac{2 \cdot 19}{2 \cdot 3} + \frac{- 10}{3}$

Step 3.8.2.1.2.3

Rewrite the expression.

$f (5) = \frac{19}{3} + \frac{- 10}{3}$

Step 3.8.2.2

Move the negative in front of the fraction.

$f (5) = \frac{19}{3} - \frac{10}{3}$

Step 3.8.3

Combine fractions.

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Step 3.8.3.1

Combine the numerators over the common denominator.

$f (5) = \frac{19 - 10}{3}$

Step 3.8.3.2

Simplify the expression.

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Step 3.8.3.2.1

Subtract $10$ from $19$ .

$f (5) = \frac{9}{3}$

Step 3.8.3.2.2

Divide $9$ by $3$ .

$f (5) = 3$

Step 3.8.4

The final answer is $3$ .

$3$

Step 3.9

The $y$ value at $x = 5$ is $3$ .

$y = 3$

Step 3.10

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

$f (3) = \frac{{(3)}^{2}}{6} - \frac{2 (3)}{3} + \frac{13}{6}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine fractions.

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Step 3.11.1.1

Combine the numerators over the common denominator.

$f (3) = \frac{{(3)}^{2} + 13}{6} + \frac{- 2 \cdot 3}{3}$

Step 3.11.1.2

Simplify the expression.

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Step 3.11.1.2.1

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

$f (3) = \frac{9 + 13}{6} + \frac{- 2 \cdot 3}{3}$

Step 3.11.1.2.2

Add $9$ and $13$ .

$f (3) = \frac{22}{6} + \frac{- 2 \cdot 3}{3}$

Step 3.11.1.2.3

Multiply $- 2$ by $3$ .

$f (3) = \frac{22}{6} + \frac{- 6}{3}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Cancel the common factor of $22$ and $6$ .

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Step 3.11.2.1.1

Factor $2$ out of $22$ .

$f (3) = \frac{2 (11)}{6} + \frac{- 6}{3}$

Step 3.11.2.1.2

Cancel the common factors.

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Step 3.11.2.1.2.1

Factor $2$ out of $6$ .

$f (3) = \frac{2 \cdot 11}{2 \cdot 3} + \frac{- 6}{3}$

Step 3.11.2.1.2.2

Cancel the common factor.

$f (3) = \frac{2 \cdot 11}{2 \cdot 3} + \frac{- 6}{3}$

Step 3.11.2.1.2.3

Rewrite the expression.

$f (3) = \frac{11}{3} + \frac{- 6}{3}$

Step 3.11.2.2

Divide $- 6$ by $3$ .

$f (3) = \frac{11}{3} - 2$

Step 3.11.3

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (3) = \frac{11}{3} - 2 \cdot \frac{3}{3}$

Step 3.11.4

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

$f (3) = \frac{11}{3} + \frac{- 2 \cdot 3}{3}$

Step 3.11.5

Combine the numerators over the common denominator.

$f (3) = \frac{11 - 2 \cdot 3}{3}$

Step 3.11.6

Simplify the numerator.

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Step 3.11.6.1

Multiply $- 2$ by $3$ .

$f (3) = \frac{11 - 6}{3}$

Step 3.11.6.2

Subtract $6$ from $11$ .

$f (3) = \frac{5}{3}$

Step 3.11.7

The final answer is $\frac{5}{3}$ .

$\frac{5}{3}$

Step 3.12

The $y$ value at $x = 3$ is $\frac{5}{3}$ .

$y = \frac{5}{3}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & \frac{13}{6} \\ 1 & \frac{5}{3} \\ 2 & \frac{3}{2} \\ 3 & \frac{5}{3} \\ 5 & 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

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

Focus: $(2, 3)$

Axis of Symmetry: $x = 2$

Directrix: $y = 0$

$\begin{array}{c} x & y \\ 0 & \frac{13}{6} \\ 1 & \frac{5}{3} \\ 2 & \frac{3}{2} \\ 3 & \frac{5}{3} \\ 5 & 3 \end{array}$

Step 5