Algebra Examples

$f (x) = \frac{- 2 x^{2} + 32}{3 x - 12}$

Step 1

Rewrite the function as an equation.

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

Step 2

Rewrite in slope-intercept form.

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Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Write in $y = m x + b$ form.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Multiply $2$ by $4$ .

$y = - \frac{8 + 2 x}{3}$

Step 2.2.3

Split the fraction $\frac{8 + 2 x}{3}$ into two fractions.

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

Step 2.2.4

Combine the numerators over the common denominator.

$y = - \frac{8 + 2 x}{3}$

Step 2.2.5

Reorder $8$ and $2 x$ .

$y = - \frac{2 x + 8}{3}$

Step 2.2.6

Factor $2$ out of $2 x + 8$ .

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Step 2.2.6.1

Factor $2$ out of $2 x$ .

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

Step 2.2.6.2

Factor $2$ out of $8$ .

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

Step 2.2.6.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

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

Step 2.2.7

Reorder terms.

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

Step 2.2.8

Remove parentheses.

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

Step 3

Use the slope-intercept form to find the slope and y-intercept.

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Step 3.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

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

$b = 0$

Step 3.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- \frac{2}{3}$

y-intercept: $(0, 0)$

Slope: $- \frac{2}{3}$

y-intercept: $(0, 0)$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 4.1

Write in $y = m x + b$ form.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $2$ by $4$ .

$y = - \frac{8 + 2 x}{3}$

Step 4.1.3

Split the fraction $\frac{8 + 2 x}{3}$ into two fractions.

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

Step 4.1.4

Combine the numerators over the common denominator.

$y = - \frac{8 + 2 x}{3}$

Step 4.1.5

Reorder $8$ and $2 x$ .

$y = - \frac{2 x + 8}{3}$

Step 4.1.6

Factor $2$ out of $2 x + 8$ .

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Step 4.1.6.1

Factor $2$ out of $2 x$ .

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

Step 4.1.6.2

Factor $2$ out of $8$ .

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

Step 4.1.6.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

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

Step 4.1.7

Reorder terms.

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

Step 4.1.8

Remove parentheses.

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

Step 4.2

Find the x-intercept.

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Step 4.2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as $- \frac{2}{3} \cdot (x + 4) = 0$ .

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

Step 4.2.2.2

Multiply both sides of the equation by $- \frac{3}{2}$ .

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

Step 4.2.2.3

Simplify both sides of the equation.

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Step 4.2.2.3.1

Simplify the left side.

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Step 4.2.2.3.1.1

Simplify $- \frac{3}{2} (- \frac{2}{3} \cdot (x + 4))$ .

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Step 4.2.2.3.1.1.1

Apply the distributive property.

$- \frac{3}{2} (- \frac{2}{3} x - \frac{2}{3} \cdot 4) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.2

Combine $x$ and $\frac{2}{3}$ .

$- \frac{3}{2} (- \frac{x \cdot 2}{3} - \frac{2}{3} \cdot 4) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.3

Multiply $- \frac{2}{3} \cdot 4$ .

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Step 4.2.2.3.1.1.3.1

Multiply $4$ by $- 1$ .

$- \frac{3}{2} (- \frac{x \cdot 2}{3} - 4 (\frac{2}{3})) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.3.2

Combine $- 4$ and $\frac{2}{3}$ .

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

Step 4.2.2.3.1.1.3.3

Multiply $- 4$ by $2$ .

$- \frac{3}{2} (- \frac{x \cdot 2}{3} + \frac{- 8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.4

Simplify each term.

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Step 4.2.2.3.1.1.4.1

Move $2$ to the left of $x$ .

$- \frac{3}{2} (- \frac{2 \cdot x}{3} + \frac{- 8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.4.2

Move the negative in front of the fraction.

$- \frac{3}{2} (- \frac{2 x}{3} - \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5

Simplify terms.

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Step 4.2.2.3.1.1.5.1

Apply the distributive property.

$- \frac{3}{2} (- \frac{2 x}{3}) - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.2

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.5.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 x}{3}) - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.2.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.2.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.2.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.2.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 x) - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.3

Cancel the common factor of $2$ .

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Step 4.2.2.3.1.1.5.3.1

Factor $2$ out of $- 2 x$ .

$\frac{- 1}{2} (2 (- x)) - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.3.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- x)) - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.3.3

Rewrite the expression.

$- - x - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.4

Multiply.

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Step 4.2.2.3.1.1.5.4.1

Multiply $- 1$ by $- 1$ .

$1 x - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.4.2

Multiply $x$ by $1$ .

$x - \frac{3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.5

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.5.5.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x + \frac{- 3}{2} (- \frac{8}{3}) = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.5.2

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$x + \frac{- 3}{2} \cdot \frac{- 8}{3} = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.5.3

Factor $3$ out of $- 3$ .

$x + \frac{3 (- 1)}{2} \cdot \frac{- 8}{3} = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.5.4

Cancel the common factor.

$x + \frac{3 \cdot - 1}{2} \cdot \frac{- 8}{3} = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.5.5

Rewrite the expression.

$x + \frac{- 1}{2} \cdot - 8 = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.6

Cancel the common factor of $2$ .

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Step 4.2.2.3.1.1.5.6.1

Factor $2$ out of $- 8$ .

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

Step 4.2.2.3.1.1.5.6.2

Cancel the common factor.

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

Step 4.2.2.3.1.1.5.6.3

Rewrite the expression.

$x - - 4 = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.1.1.5.7

Multiply $- 1$ by $- 4$ .

$x + 4 = - \frac{3}{2} \cdot 0$

Step 4.2.2.3.2

Simplify the right side.

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Step 4.2.2.3.2.1

Multiply $- \frac{3}{2} \cdot 0$ .

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Step 4.2.2.3.2.1.1

Multiply $0$ by $- 1$ .

$x + 4 = 0 (\frac{3}{2})$

Step 4.2.2.3.2.1.2

Multiply $0$ by $\frac{3}{2}$ .

$x + 4 = 0$

Step 4.2.2.4

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s): $(- 4, 0)$

Step 4.3

Find the y-intercept.

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Step 4.3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Remove parentheses.

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

Step 4.3.2.2

Remove parentheses.

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

Step 4.3.2.3

Simplify $- \frac{2}{3} \cdot ((0) + 4)$ .

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Step 4.3.2.3.1

Add $0$ and $4$ .

$y = - \frac{2}{3} \cdot 4$

Step 4.3.2.3.2

Multiply $- \frac{2}{3} \cdot 4$ .

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Step 4.3.2.3.2.1

Multiply $4$ by $- 1$ .

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

Step 4.3.2.3.2.2

Combine $- 4$ and $\frac{2}{3}$ .

$y = \frac{- 4 \cdot 2}{3}$

Step 4.3.2.3.2.3

Multiply $- 4$ by $2$ .

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

Step 4.3.2.3.3

Move the negative in front of the fraction.

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

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{8}{3})$

Step 4.4

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ - 4 & 0 \\ 0 & - \frac{8}{3} \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $- \frac{2}{3}$

y-intercept: $(0, 0)$

$\begin{array}{c} x & y \\ - 4 & 0 \\ 0 & - \frac{8}{3} \end{array}$

Step 6