Precalculus Examples

$y - 5 = - \frac{1}{2} \cdot {(x - 3)}^{2}$

Step 1

Solve for $y$ .

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Step 1.1

Simplify $- \frac{1}{2} \cdot {(x - 3)}^{2}$ .

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Step 1.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{1}{2} \cdot {(x - 3)}^{2}$

Step 1.1.2

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y - 5 = - \frac{1}{2} \cdot ((x - 3) (x - 3))$

Step 1.1.3

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

$y - 5 = - \frac{1}{2} \cdot (x (x - 3) - 3 (x - 3))$

Step 1.1.3.2

Apply the distributive property.

$y - 5 = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Step 1.1.3.3

Apply the distributive property.

$y - 5 = - \frac{1}{2} \cdot (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Multiply $x$ by $x$ .

$y - 5 = - \frac{1}{2} \cdot (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.4.1.2

Move $- 3$ to the left of $x$ .

$y - 5 = - \frac{1}{2} \cdot (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Step 1.1.4.1.3

Multiply $- 3$ by $- 3$ .

$y - 5 = - \frac{1}{2} \cdot (x^{2} - 3 x - 3 x + 9)$

Step 1.1.4.2

Subtract $3 x$ from $- 3 x$ .

$y - 5 = - \frac{1}{2} \cdot (x^{2} - 6 x + 9)$

Step 1.1.5

Apply the distributive property.

$y - 5 = - \frac{1}{2} x^{2} - \frac{1}{2} (- 6 x) - \frac{1}{2} \cdot 9$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Combine $x^{2}$ and $\frac{1}{2}$ .

$y - 5 = - \frac{x^{2}}{2} - \frac{1}{2} (- 6 x) - \frac{1}{2} \cdot 9$

Step 1.1.6.2

Cancel the common factor of $2$ .

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Step 1.1.6.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y - 5 = - \frac{x^{2}}{2} + \frac{- 1}{2} (- 6 x) - \frac{1}{2} \cdot 9$

Step 1.1.6.2.2

Factor $2$ out of $- 6 x$ .

$y - 5 = - \frac{x^{2}}{2} + \frac{- 1}{2} (2 (- 3 x)) - \frac{1}{2} \cdot 9$

Step 1.1.6.2.3

Cancel the common factor.

$y - 5 = - \frac{x^{2}}{2} + \frac{- 1}{2} (2 (- 3 x)) - \frac{1}{2} \cdot 9$

Step 1.1.6.2.4

Rewrite the expression.

$y - 5 = - \frac{x^{2}}{2} - 1 (- 3 x) - \frac{1}{2} \cdot 9$

Step 1.1.6.3

Multiply $- 3$ by $- 1$ .

$y - 5 = - \frac{x^{2}}{2} + 3 x - \frac{1}{2} \cdot 9$

Step 1.1.6.4

Multiply $- \frac{1}{2} \cdot 9$ .

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Step 1.1.6.4.1

Multiply $9$ by $- 1$ .

$y - 5 = - \frac{x^{2}}{2} + 3 x - 9 (\frac{1}{2})$

Step 1.1.6.4.2

Combine $- 9$ and $\frac{1}{2}$ .

$y - 5 = - \frac{x^{2}}{2} + 3 x + \frac{- 9}{2}$

Step 1.1.7

Move the negative in front of the fraction.

$y - 5 = - \frac{x^{2}}{2} + 3 x - \frac{9}{2}$

Step 1.2

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

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Step 1.2.1

Add $5$ to both sides of the equation.

$y = - \frac{x^{2}}{2} + 3 x - \frac{9}{2} + 5$

Step 1.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x^{2}}{2} + 3 x - \frac{9}{2} + 5 \cdot \frac{2}{2}$

Step 1.2.3

Combine $5$ and $\frac{2}{2}$ .

$y = - \frac{x^{2}}{2} + 3 x - \frac{9}{2} + \frac{5 \cdot 2}{2}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = - \frac{x^{2}}{2} + 3 x + \frac{- 9 + 5 \cdot 2}{2}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $5$ by $2$ .

$y = - \frac{x^{2}}{2} + 3 x + \frac{- 9 + 10}{2}$

Step 1.2.5.2

Add $- 9$ and $10$ .

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

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 $- \frac{x^{2}}{2} + 3 x + \frac{1}{2}$ .

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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 $c$ .

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

$b = 3$

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

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Simplify the denominator.

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Step 2.1.1.3.2.1.1

Multiply $- 1$ by $2$ .

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

Step 2.1.1.3.2.1.2

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

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

Step 2.1.1.3.2.2

Divide $- 2$ by $2$ .

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

Step 2.1.1.3.2.3

Divide $3$ by $- 1$ .

$d = - 3$

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

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

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

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

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

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 $4$ .

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

Step 2.1.1.4.2.1.2.2

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

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

Step 2.1.1.4.2.1.3

Divide $- 4$ by $2$ .

$e = \frac{1}{2} - \frac{9}{- 2}$

Step 2.1.1.4.2.1.4

Move the negative in front of the fraction.

$e = \frac{1}{2} - - \frac{9}{2}$

Step 2.1.1.4.2.1.5

Multiply $- - \frac{9}{2}$ .

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Step 2.1.1.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.4.2.1.5.2

Multiply $\frac{9}{2}$ by $1$ .

$e = \frac{1}{2} + \frac{9}{2}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 + 9}{2}$

Step 2.1.1.4.2.3

Add $1$ and $9$ .

$e = \frac{10}{2}$

Step 2.1.1.4.2.4

Divide $10$ by $2$ .

$e = 5$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 3)}^{2} + 5$ .

$- \frac{1}{2} {(x - 3)}^{2} + 5$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - 3)}^{2} + 5$

Step 2.2

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

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

$h = 3$

$k = 5$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(3, 5)$

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.

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

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 $- 1 (- 1)$ .

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

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

Step 2.5.3.2

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

$- \frac{1}{\frac{4}{2}}$

Step 2.5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(3, \frac{9}{2})$

Step 2.7

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

$x = 3$

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.

$y = \frac{11}{2}$

Step 2.9

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

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

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 $1$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

Multiply $- 1$ by $1$ .

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

Step 3.2.3

Add $- 1$ and $1$ .

$f (1) = 3 (1) + \frac{0}{2}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $3$ by $1$ .

$f (1) = 3 + \frac{0}{2}$

Step 3.2.4.2

Divide $0$ by $2$ .

$f (1) = 3 + 0$

Step 3.2.5

Add $3$ and $0$ .

$f (1) = 3$

Step 3.2.6

The final answer is $3$ .

$3$

Step 3.3

The $y$ value at $x = 1$ is $3$ .

$y = 3$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

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

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

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

Step 3.5.2.2

Multiply $- 1$ by $4$ .

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

Step 3.5.3

Add $- 4$ and $1$ .

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

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Multiply $3$ by $2$ .

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

Step 3.5.4.2

Move the negative in front of the fraction.

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

Step 3.5.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.5.6

Combine $6$ and $\frac{2}{2}$ .

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

Step 3.5.7

Combine the numerators over the common denominator.

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

Step 3.5.8

Simplify the numerator.

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Step 3.5.8.1

Multiply $6$ by $2$ .

$f (2) = \frac{12 - 3}{2}$

Step 3.5.8.2

Subtract $3$ from $12$ .

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

Step 3.5.9

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

Step 3.6

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

$y = \frac{9}{2}$

Step 3.7

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

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

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

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

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

$f (5) = 3 (5) + \frac{- 1 \cdot 25 + 1}{2}$

Step 3.8.2.2

Multiply $- 1$ by $25$ .

$f (5) = 3 (5) + \frac{- 25 + 1}{2}$

Step 3.8.3

Add $- 25$ and $1$ .

$f (5) = 3 (5) + \frac{- 24}{2}$

Step 3.8.4

Simplify each term.

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Step 3.8.4.1

Multiply $3$ by $5$ .

$f (5) = 15 + \frac{- 24}{2}$

Step 3.8.4.2

Divide $- 24$ by $2$ .

$f (5) = 15 - 12$

Step 3.8.5

Subtract $12$ from $15$ .

$f (5) = 3$

Step 3.8.6

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 $4$ in the expression.

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

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

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

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Raise $4$ to the power of $2$ .

$f (4) = 3 (4) + \frac{- 1 \cdot 16 + 1}{2}$

Step 3.11.2.2

Multiply $- 1$ by $16$ .

$f (4) = 3 (4) + \frac{- 16 + 1}{2}$

Step 3.11.3

Add $- 16$ and $1$ .

$f (4) = 3 (4) + \frac{- 15}{2}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Multiply $3$ by $4$ .

$f (4) = 12 + \frac{- 15}{2}$

Step 3.11.4.2

Move the negative in front of the fraction.

$f (4) = 12 - \frac{15}{2}$

Step 3.11.5

To write $12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (4) = 12 \cdot \frac{2}{2} - \frac{15}{2}$

Step 3.11.6

Combine $12$ and $\frac{2}{2}$ .

$f (4) = \frac{12 \cdot 2}{2} - \frac{15}{2}$

Step 3.11.7

Combine the numerators over the common denominator.

$f (4) = \frac{12 \cdot 2 - 15}{2}$

Step 3.11.8

Simplify the numerator.

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Step 3.11.8.1

Multiply $12$ by $2$ .

$f (4) = \frac{24 - 15}{2}$

Step 3.11.8.2

Subtract $15$ from $24$ .

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

Step 3.11.9

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

Step 3.12

The $y$ value at $x = 4$ is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 3 \\ 2 & \frac{9}{2} \\ 3 & 5 \\ 4 & \frac{9}{2} \\ 5 & 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(3, 5)$

Focus: $(3, \frac{9}{2})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{11}{2}$

$\begin{array}{c} x & y \\ 1 & 3 \\ 2 & \frac{9}{2} \\ 3 & 5 \\ 4 & \frac{9}{2} \\ 5 & 3 \end{array}$

Step 5