Precalculus Examples

$x^{2} = 3 y$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $3 y = x^{2}$ .

$3 y = x^{2}$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

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

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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 $c$ .

$a = \frac{1}{3}$

$b = 0$

$c = 0$

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

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

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 $0$ .

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

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.

$d = \frac{2 \cdot 0}{2 (\frac{1}{3})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{1}{3}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 3$

Step 2.1.1.3.2.3

Multiply $0$ by $3$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (\frac{1}{3})}$

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 $0$ .

$e = 0 - \frac{0}{4 (\frac{1}{3})}$

Step 2.1.1.4.2.1.2

Combine $4$ and $\frac{1}{3}$ .

$e = 0 - \frac{0}{\frac{4}{3}}$

Step 2.1.1.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (\frac{3}{4}))$

Step 2.1.1.4.2.1.4

Multiply $0$ by $\frac{3}{4}$ .

$e = 0 - 0$

Step 2.1.1.4.2.1.5

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

$\frac{1}{3} x^{2}$

Step 2.1.2

Set $y$ equal to the new right side.

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

Step 2.2

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

$a = \frac{1}{3}$

$h = 0$

$k = 0$

Step 2.3

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

Opens Up

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.

$\frac{1}{4 \cdot \frac{1}{3}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{1}{\frac{4}{3}}$

Step 2.5.3.2

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{4})$

Step 2.5.3.3

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

$\frac{3}{4}$

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.

$(0, \frac{3}{4})$

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.

$y = - \frac{3}{4}$

Step 2.9

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

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{3}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{3}{4}$

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{3}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{3}{4}$

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.

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

Step 3.2

Simplify the result.

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Step 3.2.1

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

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

Step 3.2.2

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 3.3

The $y$ value at $x = - 1$ is $\frac{1}{3}$ .

$y = \frac{1}{3}$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

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

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

Step 3.5.2

The final answer is $\frac{4}{3}$ .

$\frac{4}{3}$

Step 3.6

The $y$ value at $x = - 2$ is $\frac{4}{3}$ .

$y = \frac{4}{3}$

Step 3.7

Replace the variable $x$ with $3$ in the expression.

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

Step 3.8

Simplify the result.

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Step 3.8.1

Cancel the common factor of ${(3)}^{2}$ and $3$ .

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Step 3.8.1.1

Factor $3$ out of ${(3)}^{2}$ .

$f (3) = \frac{3 \cdot 3}{3}$

Step 3.8.1.2

Cancel the common factors.

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Step 3.8.1.2.1

Factor $3$ out of $3$ .

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

Step 3.8.1.2.2

Cancel the common factor.

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

Step 3.8.1.2.3

Rewrite the expression.

$f (3) = \frac{3}{1}$

Step 3.8.1.2.4

Divide $3$ by $1$ .

$f (3) = 3$

Step 3.8.2

The final answer is $3$ .

$3$

Step 3.9

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

$y = 3$

Step 3.10

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

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

Step 3.11

Simplify the result.

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Step 3.11.1

One to any power is one.

$f (1) = \frac{1}{3}$

Step 3.11.2

The final answer is $\frac{1}{3}$ .

$\frac{1}{3}$

Step 3.12

The $y$ value at $x = 1$ is $\frac{1}{3}$ .

$y = \frac{1}{3}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & \frac{4}{3} \\ - 1 & \frac{1}{3} \\ 0 & 0 \\ 1 & \frac{1}{3} \\ 3 & 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(0, 0)$

Focus: $(0, \frac{3}{4})$

Axis of Symmetry: $x = 0$

Directrix: $y = - \frac{3}{4}$

$\begin{array}{c} x & y \\ - 2 & \frac{4}{3} \\ - 1 & \frac{1}{3} \\ 0 & 0 \\ 1 & \frac{1}{3} \\ 3 & 3 \end{array}$

Step 5