Precalculus Examples

$x^{2} = - 24 y$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $- 24 y = x^{2}$ .

$- 24 y = x^{2}$

Step 1.2

Divide each term in $- 24 y = x^{2}$ by $- 24$ and simplify.

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Step 1.2.1

Divide each term in $- 24 y = x^{2}$ by $- 24$ .

$\frac{- 24 y}{- 24} = \frac{x^{2}}{- 24}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 24$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 24 y}{- 24} = \frac{x^{2}}{- 24}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 24}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{24}$

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 $- \frac{x^{2}}{24}$ .

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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 $c$ .

$a = - \frac{1}{24}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (- \frac{1}{24})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 2.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{24})}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{24})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{1}{24}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 24)$

Step 2.1.1.3.2.3

Multiply $0 (- 1 \cdot 24)$ .

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Step 2.1.1.3.2.3.1

Multiply $- 1$ by $24$ .

$d = 0 \cdot - 24$

Step 2.1.1.3.2.3.2

Multiply $0$ by $- 24$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (- \frac{1}{24})}$

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

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

$e = 0 - \frac{0}{4 (- \frac{1}{24})}$

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 $4$ .

$e = 0 - \frac{0}{- 4 (\frac{1}{24})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{24}$ .

$e = 0 - \frac{0}{\frac{- 4}{24}}$

Step 2.1.1.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 2.1.1.4.2.1.3.1

Cancel the common factor of $- 4$ and $24$ .

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Step 2.1.1.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = 0 - \frac{0}{\frac{4 (- 1)}{24}}$

Step 2.1.1.4.2.1.3.1.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.1.2.1

Factor $4$ out of $24$ .

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Step 2.1.1.4.2.1.3.1.2.2

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Step 2.1.1.4.2.1.3.1.2.3

Rewrite the expression.

$e = 0 - \frac{0}{\frac{- 1}{6}}$

Step 2.1.1.4.2.1.3.2

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{1}{6}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- 1 \cdot 6))$

Step 2.1.1.4.2.1.5

Multiply $- (0 (- 1 \cdot 6))$ .

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Step 2.1.1.4.2.1.5.1

Multiply $- 1$ by $6$ .

$e = 0 - (0 \cdot - 6)$

Step 2.1.1.4.2.1.5.2

Multiply $0$ by $- 6$ .

$e = 0 - 0$

Step 2.1.1.4.2.1.5.3

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 2.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{24} x^{2}$ .

$- \frac{1}{24} x^{2}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{24} x^{2}$

Step 2.2

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

$a = - \frac{1}{24}$

$h = 0$

$k = 0$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(0, 0)$

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.

$\frac{1}{4 \cdot (- \frac{1}{24})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{24})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{24})}$

Step 2.5.3.2

Combine $4$ and $\frac{1}{24}$ .

$- \frac{1}{\frac{4}{24}}$

Step 2.5.3.3

Cancel the common factor of $4$ and $24$ .

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Step 2.5.3.3.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{24}}$

Step 2.5.3.3.2

Cancel the common factors.

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Step 2.5.3.3.2.1

Factor $4$ out of $24$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 2.5.3.3.2.2

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 2.5.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{1}{6}}$

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 6)$

Step 2.5.3.5

Multiply $- (1 \cdot 6)$ .

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Step 2.5.3.5.1

Multiply $6$ by $1$ .

$- 1 \cdot 6$

Step 2.5.3.5.2

Multiply $- 1$ by $6$ .

$- 6$

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.

$(0, - 6)$

Step 2.7

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

$x = 0$

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.

$y = 6$

Step 2.9

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

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - 6)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - 6)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

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.

$f (- 1) = - \frac{{(- 1)}^{2}}{24}$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (- 1) = - \frac{1}{24}$

Step 3.2.2

The final answer is $- \frac{1}{24}$ .

$- \frac{1}{24}$

Step 3.3

The $y$ value at $x = - 1$ is $- \frac{1}{24}$ .

$y = - \frac{1}{24}$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

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

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

Step 3.5.2

Cancel the common factor of $4$ and $24$ .

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Step 3.5.2.1

Factor $4$ out of $4$ .

$f (- 2) = - \frac{4 (1)}{24}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $4$ out of $24$ .

$f (- 2) = - \frac{4 \cdot 1}{4 \cdot 6}$

Step 3.5.2.2.2

Cancel the common factor.

$f (- 2) = - \frac{4 \cdot 1}{4 \cdot 6}$

Step 3.5.2.2.3

Rewrite the expression.

$f (- 2) = - \frac{1}{6}$

Step 3.5.3

The final answer is $- \frac{1}{6}$ .

$- \frac{1}{6}$

Step 3.6

The $y$ value at $x = - 2$ is $- \frac{1}{6}$ .

$y = - \frac{1}{6}$

Step 3.7

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

$f (1) = - \frac{{(1)}^{2}}{24}$

Step 3.8

Simplify the result.

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Step 3.8.1

One to any power is one.

$f (1) = - \frac{1}{24}$

Step 3.8.2

The final answer is $- \frac{1}{24}$ .

$- \frac{1}{24}$

Step 3.9

The $y$ value at $x = 1$ is $- \frac{1}{24}$ .

$y = - \frac{1}{24}$

Step 3.10

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

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

Step 3.11

Simplify the result.

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Step 3.11.1

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

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

Step 3.11.2

Cancel the common factor of $4$ and $24$ .

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Step 3.11.2.1

Factor $4$ out of $4$ .

$f (2) = - \frac{4 (1)}{24}$

Step 3.11.2.2

Cancel the common factors.

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Step 3.11.2.2.1

Factor $4$ out of $24$ .

$f (2) = - \frac{4 \cdot 1}{4 \cdot 6}$

Step 3.11.2.2.2

Cancel the common factor.

$f (2) = - \frac{4 \cdot 1}{4 \cdot 6}$

Step 3.11.2.2.3

Rewrite the expression.

$f (2) = - \frac{1}{6}$

Step 3.11.3

The final answer is $- \frac{1}{6}$ .

$- \frac{1}{6}$

Step 3.12

The $y$ value at $x = 2$ is $- \frac{1}{6}$ .

$y = - \frac{1}{6}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - \frac{1}{6} \\ - 1 & - \frac{1}{24} \\ 0 & 0 \\ 1 & - \frac{1}{24} \\ 2 & - \frac{1}{6} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - 6)$

Axis of Symmetry: $x = 0$

Directrix: $y = 6$

$\begin{array}{c} x & y \\ - 2 & - \frac{1}{6} \\ - 1 & - \frac{1}{24} \\ 0 & 0 \\ 1 & - \frac{1}{24} \\ 2 & - \frac{1}{6} \end{array}$

Step 5