Precalculus Examples

$y^{2} - 3 y - x + 4 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

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

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Step 1.1.1.1

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

$- 3 y - x + 4 = - y^{2}$

Step 1.1.1.2

Add $3 y$ to both sides of the equation.

$- x + 4 = - y^{2} + 3 y$

Step 1.1.1.3

Subtract $4$ from both sides of the equation.

$- x = - y^{2} + 3 y - 4$

Step 1.1.2

Divide each term in $- x = - y^{2} + 3 y - 4$ by $- 1$ and simplify.

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Step 1.1.2.1

Divide each term in $- x = - y^{2} + 3 y - 4$ by $- 1$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.2.2

Divide $x$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.2

Divide $y^{2}$ by $1$ .

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

Step 1.1.2.3.1.3

Move the negative one from the denominator of $\frac{3 y}{- 1}$ .

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

Step 1.1.2.3.1.4

Rewrite $- 1 \cdot (3 y)$ as $- (3 y)$ .

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

Step 1.1.2.3.1.5

Multiply $3$ by $- 1$ .

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

Step 1.1.2.3.1.6

Divide $- 4$ by $- 1$ .

$x = y^{2} - 3 y + 4$

Step 1.2

Complete the square for $y^{2} - 3 y + 4$ .

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Step 1.2.1

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

$a = 1$

$b = - 3$

$c = 4$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $1$ .

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

Step 1.2.3.2.2

Move the negative in front of the fraction.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

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

Step 1.2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.2.4.2.2

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

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

Step 1.2.4.2.3

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

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

Step 1.2.4.2.4

Combine the numerators over the common denominator.

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

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $4$ by $4$ .

$e = \frac{16 - 9}{4}$

Step 1.2.4.2.5.2

Subtract $9$ from $16$ .

$e = \frac{7}{4}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{3}{2})}^{2} + \frac{7}{4}$ .

${(y - \frac{3}{2})}^{2} + \frac{7}{4}$

Step 1.3

Set $x$ equal to the new right side.

$x = {(y - \frac{3}{2})}^{2} + \frac{7}{4}$

Step 2

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

$a = 1$

$h = \frac{7}{4}$

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

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(\frac{7}{4}, \frac{3}{2})$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 6

Find the focus.

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Step 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 6.2

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

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

Step 7

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

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

Step 8

Find the directrix.

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Step 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 8.2

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

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

Step 9

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

Direction: Opens Right

Vertex: $(\frac{7}{4}, \frac{3}{2})$

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

Axis of Symmetry: $y = \frac{3}{2}$

Directrix: $x = \frac{3}{2}$

Step 10