Pre-Algebra Examples

$x^{2} - 2 x + 4 y - 5 = 0$

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.

$- 2 x + 4 y - 5 = - x^{2}$

Step 1.1.2

Add $2 x$ to both sides of the equation.

$4 y - 5 = - x^{2} + 2 x$

Step 1.1.3

Add $5$ to both sides of the equation.

$4 y = - x^{2} + 2 x + 5$

Step 1.2

Divide each term in $4 y = - x^{2} + 2 x + 5$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 y = - x^{2} + 2 x + 5$ by $4$ .

$\frac{4 y}{4} = \frac{- x^{2}}{4} + \frac{2 x}{4} + \frac{5}{4}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- x^{2}}{4} + \frac{2 x}{4} + \frac{5}{4}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{4} + \frac{2 x}{4} + \frac{5}{4}$

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

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} + \frac{2 x}{4} + \frac{5}{4}$

Step 1.2.3.1.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $2 x$ .

$y = - \frac{x^{2}}{4} + \frac{2 (x)}{4} + \frac{5}{4}$

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 $4$ .

$y = - \frac{x^{2}}{4} + \frac{2 x}{2 \cdot 2} + \frac{5}{4}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{4} + \frac{2 x}{2 \cdot 2} + \frac{5}{4}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{4} + \frac{x}{2} + \frac{5}{4}$

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 $- \frac{x^{2}}{4} + \frac{x}{2} + \frac{5}{4}$ .

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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 $c$ .

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

$b = \frac{1}{2}$

$c = \frac{5}{4}$

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

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

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.

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

Step 2.1.1.3.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 2.1.1.3.2.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 2.1.1.3.2.2.2

Move the negative in front of the fraction.

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

Step 2.1.1.3.2.3

Combine $2$ and $\frac{1}{4}$ .

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

Step 2.1.1.3.2.4

Cancel the common factor of $2$ and $4$ .

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Step 2.1.1.3.2.4.1

Factor $2$ out of $2$ .

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

Step 2.1.1.3.2.4.2

Cancel the common factors.

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Step 2.1.1.3.2.4.2.1

Factor $2$ out of $4$ .

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

Step 2.1.1.3.2.4.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.4.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.6

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.6.1

Factor $2$ out of $- (1 \cdot 2)$ .

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

Step 2.1.1.3.2.6.2

Cancel the common factor.

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

Step 2.1.1.3.2.6.3

Rewrite the expression.

$d = - 1 \cdot (1)$

Step 2.1.1.3.2.7

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = \frac{5}{4} - \frac{{(\frac{1}{2})}^{2}}{4 (- \frac{1}{4})}$

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 $\frac{1}{2}$ .

$e = \frac{5}{4} - \frac{\frac{1^{2}}{2^{2}}}{4 (- \frac{1}{4})}$

Step 2.1.1.4.2.1.1.2

One to any power is one.

$e = \frac{5}{4} - \frac{\frac{1}{2^{2}}}{4 (- \frac{1}{4})}$

Step 2.1.1.4.2.1.1.3

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

$e = \frac{5}{4} - \frac{\frac{1}{4}}{4 (- \frac{1}{4})}$

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{5}{4} - \frac{\frac{1}{4}}{- 4 (\frac{1}{4})}$

Step 2.1.1.4.2.1.2.2

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

$e = \frac{5}{4} - \frac{\frac{1}{4}}{\frac{- 4}{4}}$

Step 2.1.1.4.2.1.3

Divide $- 4$ by $4$ .

$e = \frac{5}{4} - \frac{\frac{1}{4}}{- 1}$

Step 2.1.1.4.2.1.4

Move the negative one from the denominator of $\frac{\frac{1}{4}}{- 1}$ .

$e = \frac{5}{4} - (- 1 \cdot \frac{1}{4})$

Step 2.1.1.4.2.1.5

Rewrite $- 1 \cdot \frac{1}{4}$ as $- \frac{1}{4}$ .

$e = \frac{5}{4} - - \frac{1}{4}$

Step 2.1.1.4.2.1.6

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

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Step 2.1.1.4.2.1.6.1

Multiply $- 1$ by $- 1$ .

$e = \frac{5}{4} + 1 (\frac{1}{4})$

Step 2.1.1.4.2.1.6.2

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

$e = \frac{5}{4} + \frac{1}{4}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{5 + 1}{4}$

Step 2.1.1.4.2.3

Add $5$ and $1$ .

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

Step 2.1.1.4.2.4

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

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Step 2.1.1.4.2.4.1

Factor $2$ out of $6$ .

$e = \frac{2 (3)}{4}$

Step 2.1.1.4.2.4.2

Cancel the common factors.

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Step 2.1.1.4.2.4.2.1

Factor $2$ out of $4$ .

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

Step 2.1.1.4.2.4.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.4.2.3

Rewrite the expression.

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

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x - 1)}^{2} + \frac{3}{2}$ .

$- \frac{1}{4} {(x - 1)}^{2} + \frac{3}{2}$

Step 2.1.2

Set $y$ equal to the new right side.

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

Step 2.2

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

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

$h = 1$

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

Step 2.3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

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

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.

$\frac{1}{4 \cdot (- \frac{1}{4})}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{4})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

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

Step 2.5.3.2

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

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

Step 2.5.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 2.5.3.4

Cancel the common factor of $1$ .

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Step 2.5.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 2.5.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 2.5.3.5

Multiply $- 1$ by $1$ .

$- 1$

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.

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

Step 2.7

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

$x = 1$

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.

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

Step 2.9

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

Direction: Opens Down

Vertex: $(1, \frac{3}{2})$

Focus: $(1, \frac{1}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{5}{2}$

Direction: Opens Down

Vertex: $(1, \frac{3}{2})$

Focus: $(1, \frac{1}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{5}{2}$

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 $0$ in the expression.

$f (0) = - \frac{{(0)}^{2}}{4} + \frac{0}{2} + \frac{5}{4}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (0) = \frac{- {(0)}^{2} + 5}{4} + \frac{0}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (0) = \frac{- 0 + 5}{4} + \frac{0}{2}$

Step 3.2.2.2

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 + 5}{4} + \frac{0}{2}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Add $0$ and $5$ .

$f (0) = \frac{5}{4} + \frac{0}{2}$

Step 3.2.3.2

Divide $0$ by $2$ .

$f (0) = \frac{5}{4} + 0$

Step 3.2.3.3

Add $\frac{5}{4}$ and $0$ .

$f (0) = \frac{5}{4}$

Step 3.2.4

The final answer is $\frac{5}{4}$ .

$\frac{5}{4}$

Step 3.3

The $y$ value at $x = 0$ is $\frac{5}{4}$ .

$y = \frac{5}{4}$

Step 3.4

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

$f (- 1) = - \frac{{(- 1)}^{2}}{4} + \frac{- 1}{2} + \frac{5}{4}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (- 1) = \frac{- {(- 1)}^{2} + 5}{4} + \frac{- 1}{2}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.5.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.5.2.1.1.1

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

$f (- 1) = \frac{(- 1) {(- 1)}^{2} + 5}{4} + \frac{- 1}{2}$

Step 3.5.2.1.1.2

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

$f (- 1) = \frac{{(- 1)}^{1 + 2} + 5}{4} + \frac{- 1}{2}$

Step 3.5.2.1.2

Add $1$ and $2$ .

$f (- 1) = \frac{{(- 1)}^{3} + 5}{4} + \frac{- 1}{2}$

Step 3.5.2.2

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{- 1 + 5}{4} + \frac{- 1}{2}$

Step 3.5.3

Add $- 1$ and $5$ .

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

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Divide $4$ by $4$ .

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

Step 3.5.4.2

Move the negative in front of the fraction.

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

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

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

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

Step 3.5.5.2

Combine the numerators over the common denominator.

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

Step 3.5.5.3

Subtract $1$ from $2$ .

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

Step 3.5.6

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3.6

The $y$ value at $x = - 1$ is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 3.7

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

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

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

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

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

$f (2) = \frac{- 1 \cdot 4 + 5}{4} + \frac{2}{2}$

Step 3.8.2.2

Multiply $- 1$ by $4$ .

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

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Add $- 4$ and $5$ .

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

Step 3.8.3.2

Divide $2$ by $2$ .

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

Step 3.8.3.3

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

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

Step 3.8.3.4

Combine the numerators over the common denominator.

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

Step 3.8.3.5

Add $1$ and $4$ .

$f (2) = \frac{5}{4}$

Step 3.8.4

The final answer is $\frac{5}{4}$ .

$\frac{5}{4}$

Step 3.9

The $y$ value at $x = 2$ is $\frac{5}{4}$ .

$y = \frac{5}{4}$

Step 3.10

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

$f (3) = - \frac{{(3)}^{2}}{4} + \frac{3}{2} + \frac{5}{4}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (3) = \frac{- {(3)}^{2} + 5}{4} + \frac{3}{2}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

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

$f (3) = \frac{- 1 \cdot 9 + 5}{4} + \frac{3}{2}$

Step 3.11.2.2

Multiply $- 1$ by $9$ .

$f (3) = \frac{- 9 + 5}{4} + \frac{3}{2}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Add $- 9$ and $5$ .

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

Step 3.11.3.2

Divide $- 4$ by $4$ .

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

Step 3.11.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.11.5

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 3.11.6

Combine the numerators over the common denominator.

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

Step 3.11.7

Simplify the numerator.

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Step 3.11.7.1

Multiply $- 1$ by $2$ .

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

Step 3.11.7.2

Add $- 2$ and $3$ .

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

Step 3.11.8

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3.12

The $y$ value at $x = 3$ is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

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

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, \frac{3}{2})$

Focus: $(1, \frac{1}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = \frac{5}{2}$

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

Step 5