Algebra Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $- \frac{x^{2}}{3} + \frac{4}{3}$ .

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Step 1.1.1.1

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

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

$b = 0$

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

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

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

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Step 1.1.1.3.2.1.1

Factor $2$ out of $0$ .

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

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

Cancel the common factor.

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

Step 1.1.1.3.2.1.2.2

Rewrite the expression.

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

Step 1.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 3)$

Step 1.1.1.3.2.3

Multiply $0 (- 1 \cdot 3)$ .

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Step 1.1.1.3.2.3.1

Multiply $- 1$ by $3$ .

$d = 0 \cdot - 3$

Step 1.1.1.3.2.3.2

Multiply $0$ by $- 3$ .

$d = 0$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

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

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

Step 1.1.1.4.2.1.2

Simplify the denominator.

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Step 1.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 1.1.1.4.2.1.2.2

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

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

Step 1.1.1.4.2.1.3

Move the negative in front of the fraction.

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

Step 1.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.1.4.2.1.5

Multiply $0 (- \frac{3}{4})$ .

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Step 1.1.1.4.2.1.5.1

Multiply $- 1$ by $0$ .

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

Step 1.1.1.4.2.1.5.2

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

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

Step 1.1.1.4.2.1.6

Multiply $- 1$ by $0$ .

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

Step 1.1.1.4.2.2

Add $\frac{4}{3}$ and $0$ .

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

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{3} {(x + 0)}^{2} + \frac{4}{3}$ .

$- \frac{1}{3} {(x + 0)}^{2} + \frac{4}{3}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{3} \cdot {(x + 0)}^{2} + \frac{4}{3}$

Step 1.2

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

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

$h = 0$

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

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

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

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 1.5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.5.3.1.2

Move the negative in front of the fraction.

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

Step 1.5.3.2

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

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

Step 1.5.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.3.4

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

$- \frac{3}{4}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(0, \frac{7}{12})$

Step 1.7

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

$x = 0$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$y = \frac{25}{12}$

Step 1.9

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

Direction: Opens Down

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

Focus: $(0, \frac{7}{12})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{25}{12}$

Direction: Opens Down

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

Focus: $(0, \frac{7}{12})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{25}{12}$

Step 2

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 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.2.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 2.2.2.1.1.1

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

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

Step 2.2.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.2.1.2

Add $1$ and $2$ .

$f (- 1) = \frac{{(- 1)}^{3} + 4}{3}$

Step 2.2.2.2

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{- 1 + 4}{3}$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Add $- 1$ and $4$ .

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

Step 2.2.3.2

Divide $3$ by $3$ .

$f (- 1) = 1$

Step 2.2.4

The final answer is $1$ .

$1$

Step 2.3

The $y$ value at $x = - 1$ is $1$ .

$y = 1$

Step 2.4

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

Combine the numerators over the common denominator.

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

Step 2.5.2

Simplify each term.

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Step 2.5.2.1

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

$f (- 2) = \frac{- 1 \cdot 4 + 4}{3}$

Step 2.5.2.2

Multiply $- 1$ by $4$ .

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

Step 2.5.3

Simplify the expression.

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Step 2.5.3.1

Add $- 4$ and $4$ .

$f (- 2) = \frac{0}{3}$

Step 2.5.3.2

Divide $0$ by $3$ .

$f (- 2) = 0$

Step 2.5.4

The final answer is $0$ .

$0$

Step 2.6

The $y$ value at $x = - 2$ is $0$ .

$y = 0$

Step 2.7

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

Combine the numerators over the common denominator.

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

Step 2.8.2

Simplify each term.

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Step 2.8.2.1

One to any power is one.

$f (1) = \frac{- 1 \cdot 1 + 4}{3}$

Step 2.8.2.2

Multiply $- 1$ by $1$ .

$f (1) = \frac{- 1 + 4}{3}$

Step 2.8.3

Simplify the expression.

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Step 2.8.3.1

Add $- 1$ and $4$ .

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

Step 2.8.3.2

Divide $3$ by $3$ .

$f (1) = 1$

Step 2.8.4

The final answer is $1$ .

$1$

Step 2.9

The $y$ value at $x = 1$ is $1$ .

$y = 1$

Step 2.10

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Combine the numerators over the common denominator.

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

Step 2.11.2

Simplify each term.

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Step 2.11.2.1

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

$f (2) = \frac{- 1 \cdot 4 + 4}{3}$

Step 2.11.2.2

Multiply $- 1$ by $4$ .

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

Step 2.11.3

Simplify the expression.

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Step 2.11.3.1

Add $- 4$ and $4$ .

$f (2) = \frac{0}{3}$

Step 2.11.3.2

Divide $0$ by $3$ .

$f (2) = 0$

Step 2.11.4

The final answer is $0$ .

$0$

Step 2.12

The $y$ value at $x = 2$ is $0$ .

$y = 0$

Step 2.13

Graph the parabola using its properties and the selected points.

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

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

Focus: $(0, \frac{7}{12})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{25}{12}$

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

Step 4