Precalculus Examples

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

Step 1

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

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Step 1.1

Rewrite.

$x - 4 = 0 + 0 + \frac{1}{2} \cdot {(y - 1)}^{2}$

Step 1.2

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

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

Step 1.3

Expand $(y - 1) (y - 1)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $y$ by $y$ .

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

Step 1.4.1.2

Move $- 1$ to the left of $y$ .

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

Step 1.4.1.3

Rewrite $- 1 y$ as $- y$ .

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

Step 1.4.1.4

Rewrite $- 1 y$ as $- y$ .

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

Step 1.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.4.2

Subtract $y$ from $- y$ .

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Simplify.

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Step 1.6.1

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

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

Step 1.6.2

Cancel the common factor of $2$ .

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Step 1.6.2.1

Factor $2$ out of $- 2 y$ .

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

Step 1.6.2.2

Cancel the common factor.

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

Step 1.6.2.3

Rewrite the expression.

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

Step 1.6.3

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

$x - 4 = \frac{y^{2}}{2} - y + \frac{1}{2}$

Step 2

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

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Step 2.1

Add $4$ to both sides of the equation.

$x = \frac{y^{2}}{2} - y + \frac{1}{2} + 4$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $4$ by $2$ .

$x = \frac{y^{2}}{2} - y + \frac{1 + 8}{2}$

Step 2.5.2

Add $1$ and $8$ .

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

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

Complete the square for $\frac{y^{2}}{2} - y + \frac{9}{2}$ .

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Step 3.1.1.1

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

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

$b = - 1$

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

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

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

Step 3.1.1.3.2

Simplify the right side.

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Step 3.1.1.3.2.1

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

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

Step 3.1.1.3.2.2

Divide $2$ by $2$ .

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

Step 3.1.1.3.2.3

Divide $- 1$ by $1$ .

$d = - 1$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

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

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

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

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

Step 3.1.1.4.2.1.2

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

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

Step 3.1.1.4.2.1.3

Divide $4$ by $2$ .

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

Step 3.1.1.4.2.2

Combine the numerators over the common denominator.

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

Step 3.1.1.4.2.3

Subtract $1$ from $9$ .

$e = \frac{8}{2}$

Step 3.1.1.4.2.4

Divide $8$ by $2$ .

$e = 4$

Step 3.1.1.5

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

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

Step 3.1.2

Set $x$ equal to the new right side.

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

Step 3.2

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

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

$h = 4$

$k = 1$

Step 3.3

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

Opens Right

Step 3.4

Find the vertex $(h, k)$ .

$(4, 1)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

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

Step 3.5.3

Simplify.

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Step 3.5.3.1

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

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

Step 3.5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 3.6

Find the focus.

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Step 3.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 3.6.2

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

$(\frac{9}{2}, 1)$

Step 3.7

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

$y = 1$

Step 3.8

Find the directrix.

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Step 3.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 3.8.2

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

$x = \frac{7}{2}$

Step 3.9

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

Direction: Opens Right

Vertex: $(4, 1)$

Focus: $(\frac{9}{2}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{7}{2}$

Direction: Opens Right

Vertex: $(4, 1)$

Focus: $(\frac{9}{2}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{7}{2}$

Step 4

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 4.1

Substitute the $x$ value $5$ into $f (x) = \sqrt{2 (x - 4)} + 1$ . In this case, the point is $(5, 2.41421356)$ .

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $4$ from $5$ .

$f (5) = \sqrt{2 \cdot 1} + 1$

Step 4.1.2.1.2

Multiply $2$ by $1$ .

$f (5) = \sqrt{2} + 1$

Step 4.1.2.2

The final answer is $\sqrt{2} + 1$ .

$y = \sqrt{2} + 1$

Step 4.1.3

Convert $\sqrt{2} + 1$ to decimal.

$= 2.41421356$

Step 4.2

Substitute the $x$ value $5$ into $f (x) = - \sqrt{2 (x - 4)} + 1$ . In this case, the point is $(5, - 0.41421356)$ .

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Subtract $4$ from $5$ .

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

Step 4.2.2.1.2

Multiply $2$ by $1$ .

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

Step 4.2.2.2

The final answer is $- \sqrt{2} + 1$ .

$y = - \sqrt{2} + 1$

Step 4.2.3

Convert $- \sqrt{2} + 1$ to decimal.

$= - 0.41421356$

Step 4.3

Substitute the $x$ value $6$ into $f (x) = \sqrt{2 (x - 4)} + 1$ . In this case, the point is $(6, 3)$ .

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Step 4.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \sqrt{2 ((6) - 4)} + 1$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Subtract $4$ from $6$ .

$f (6) = \sqrt{2 \cdot 2} + 1$

Step 4.3.2.1.2

Multiply $2$ by $2$ .

$f (6) = \sqrt{4} + 1$

Step 4.3.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (6) = \sqrt{2^{2}} + 1$

Step 4.3.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (6) = 2 + 1$

Step 4.3.2.2

Add $2$ and $1$ .

$f (6) = 3$

Step 4.3.2.3

The final answer is $3$ .

$y = 3$

Step 4.3.3

Convert $3$ to decimal.

$= 3$

Step 4.4

Substitute the $x$ value $6$ into $f (x) = - \sqrt{2 (x - 4)} + 1$ . In this case, the point is $(6, - 1)$ .

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Step 4.4.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = - \sqrt{2 ((6) - 4)} + 1$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Subtract $4$ from $6$ .

$f (6) = - \sqrt{2 \cdot 2} + 1$

Step 4.4.2.1.2

Multiply $2$ by $2$ .

$f (6) = - \sqrt{4} + 1$

Step 4.4.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (6) = - \sqrt{2^{2}} + 1$

Step 4.4.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (6) = - 1 \cdot 2 + 1$

Step 4.4.2.1.5

Multiply $- 1$ by $2$ .

$f (6) = - 2 + 1$

Step 4.4.2.2

Add $- 2$ and $1$ .

$f (6) = - 1$

Step 4.4.2.3

The final answer is $- 1$ .

$y = - 1$

Step 4.4.3

Convert $- 1$ to decimal.

$= - 1$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & 1 \\ 5 & 2.41 \\ 5 & - 0.41 \\ 6 & 3 \\ 6 & - 1 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(4, 1)$

Focus: $(\frac{9}{2}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{7}{2}$

$\begin{array}{c} x & y \\ 4 & 1 \\ 5 & 2.41 \\ 5 & - 0.41 \\ 6 & 3 \\ 6 & - 1 \end{array}$

Step 6