Precalculus Examples

$4 x^{2} + 2 y = - 3 x - 6$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Subtract $4 x^{2}$ from both sides of the equation.

$2 y = - 3 x - 6 - 4 x^{2}$

Step 1.1.2

Divide each term in $2 y = - 3 x - 6 - 4 x^{2}$ by $2$ and simplify.

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Step 1.1.2.1

Divide each term in $2 y = - 3 x - 6 - 4 x^{2}$ by $2$ .

$\frac{2 y}{2} = \frac{- 3 x}{2} + \frac{- 6}{2} + \frac{- 4 x^{2}}{2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 3 x}{2} + \frac{- 6}{2} + \frac{- 4 x^{2}}{2}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{2} + \frac{- 6}{2} + \frac{- 4 x^{2}}{2}$

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

Move the negative in front of the fraction.

$y = - \frac{3 x}{2} + \frac{- 6}{2} + \frac{- 4 x^{2}}{2}$

Step 1.1.2.3.1.2

Divide $- 6$ by $2$ .

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

Step 1.1.2.3.1.3

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

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Step 1.1.2.3.1.3.1

Factor $2$ out of $- 4 x^{2}$ .

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

Step 1.1.2.3.1.3.2

Cancel the common factors.

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Step 1.1.2.3.1.3.2.1

Factor $2$ out of $2$ .

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

Step 1.1.2.3.1.3.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.3.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.3.2.4

Divide $- 2 x^{2}$ by $1$ .

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

Step 1.2

Complete the square for $- \frac{3 x}{2} - 3 - 2 x^{2}$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Move $- 3$ .

$- \frac{3 x}{2} - 2 x^{2} - 3$

Step 1.2.1.2

Reorder $- \frac{3 x}{2}$ and $- 2 x^{2}$ .

$- 2 x^{2} - \frac{3 x}{2} - 3$

Step 1.2.2

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

$a = - 2$

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

$c = - 3$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.2

Multiply $2$ by $- 2$ .

$d = - \frac{3}{2} \cdot \frac{1}{- 4}$

Step 1.2.4.2.3

Move the negative in front of the fraction.

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

Step 1.2.4.2.4

Multiply $- \frac{3}{2} (- \frac{1}{4})$ .

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Step 1.2.4.2.4.1

Multiply $- 1$ by $- 1$ .

$d = 1 (\frac{3}{2}) \frac{1}{4}$

Step 1.2.4.2.4.2

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

$d = \frac{3}{2} \cdot \frac{1}{4}$

Step 1.2.4.2.4.3

Multiply $\frac{3}{2}$ by $\frac{1}{4}$ .

$d = \frac{3}{2 \cdot 4}$

Step 1.2.4.2.4.4

Multiply $2$ by $4$ .

$d = \frac{3}{8}$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

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

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

Step 1.2.5.2.1.1.2

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

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

Step 1.2.5.2.1.1.3

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

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

Step 1.2.5.2.1.1.4

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

$e = - 3 - \frac{1 \frac{9}{2^{2}}}{4 \cdot - 2}$

Step 1.2.5.2.1.1.5

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

$e = - 3 - \frac{1 (\frac{9}{4})}{4 \cdot - 2}$

Step 1.2.5.2.1.1.6

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

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

Step 1.2.5.2.1.2

Multiply $4$ by $- 2$ .

$e = - 3 - \frac{\frac{9}{4}}{- 8}$

Step 1.2.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = - 3 - (\frac{9}{4} \cdot \frac{1}{- 8})$

Step 1.2.5.2.1.4

Move the negative in front of the fraction.

$e = - 3 - (\frac{9}{4} (- \frac{1}{8}))$

Step 1.2.5.2.1.5

Multiply $\frac{9}{4} (- \frac{1}{8})$ .

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Step 1.2.5.2.1.5.1

Multiply $\frac{9}{4}$ by $\frac{1}{8}$ .

$e = - 3 - - \frac{9}{4 \cdot 8}$

Step 1.2.5.2.1.5.2

Multiply $4$ by $8$ .

$e = - 3 - - \frac{9}{32}$

Step 1.2.5.2.1.6

Multiply $- - \frac{9}{32}$ .

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Step 1.2.5.2.1.6.1

Multiply $- 1$ by $- 1$ .

$e = - 3 + 1 (\frac{9}{32})$

Step 1.2.5.2.1.6.2

Multiply $\frac{9}{32}$ by $1$ .

$e = - 3 + \frac{9}{32}$

Step 1.2.5.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$e = - 3 \cdot \frac{32}{32} + \frac{9}{32}$

Step 1.2.5.2.3

Combine $- 3$ and $\frac{32}{32}$ .

$e = \frac{- 3 \cdot 32}{32} + \frac{9}{32}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 3 \cdot 32 + 9}{32}$

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $- 3$ by $32$ .

$e = \frac{- 96 + 9}{32}$

Step 1.2.5.2.5.2

Add $- 96$ and $9$ .

$e = \frac{- 87}{32}$

Step 1.2.5.2.6

Move the negative in front of the fraction.

$e = - \frac{87}{32}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + \frac{3}{8})}^{2} - \frac{87}{32}$ .

$- 2 {(x + \frac{3}{8})}^{2} - \frac{87}{32}$

Step 1.3

Set $y$ equal to the new right side.

$y = - 2 {(x + \frac{3}{8})}^{2} - \frac{87}{32}$

Step 2

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

$a = - 2$

$h = - \frac{3}{8}$

$k = - \frac{87}{32}$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- \frac{3}{8}, - \frac{87}{32})$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(- \frac{3}{8}, - \frac{91}{32})$

Step 7

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

$x = - \frac{3}{8}$

Step 8

Find the directrix.

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Step 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 8.2

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

$y = - \frac{83}{32}$

Step 9

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

Direction: Opens Down

Vertex: $(- \frac{3}{8}, - \frac{87}{32})$

Focus: $(- \frac{3}{8}, - \frac{91}{32})$

Axis of Symmetry: $x = - \frac{3}{8}$

Directrix: $y = - \frac{83}{32}$

Step 10