Precalculus Examples

$f (x) = \frac{- 6 x - 9}{2}$

Step 1

Rewrite the function as an equation.

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

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.

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

Step 2.2.2

Multiply $2$ by $3$ .

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

Step 2.2.3

Multiply $3$ by $3$ .

$y = - \frac{6 x + 9}{2}$

Step 2.2.4

Split the fraction $\frac{6 x + 9}{2}$ into two fractions.

$y = - (\frac{6 x}{2} + \frac{9}{2})$

Step 2.2.5

Cancel the common factor of $6$ and $2$ .

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Step 2.2.5.1

Factor $2$ out of $6 x$ .

$y = - (\frac{2 (3 x)}{2} + \frac{9}{2})$

Step 2.2.5.2

Cancel the common factors.

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Step 2.2.5.2.1

Factor $2$ out of $2$ .

$y = - (\frac{2 (3 x)}{2 (1)} + \frac{9}{2})$

Step 2.2.5.2.2

Cancel the common factor.

$y = - (\frac{2 (3 x)}{2 \cdot 1} + \frac{9}{2})$

Step 2.2.5.2.3

Rewrite the expression.

$y = - (\frac{3 x}{1} + \frac{9}{2})$

Step 2.2.5.2.4

Divide $3 x$ by $1$ .

$y = - (3 x + \frac{9}{2})$

Step 2.2.6

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - (3 x \cdot \frac{2}{2} + \frac{9}{2})$

Step 2.2.7

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

$y = - (\frac{3 x \cdot 2}{2} + \frac{9}{2})$

Step 2.2.8

Combine the numerators over the common denominator.

$y = - \frac{3 x \cdot 2 + 9}{2}$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Factor $3$ out of $3 x \cdot 2 + 9$ .

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Step 2.2.9.1.1

Factor $3$ out of $3 x \cdot 2$ .

$y = - \frac{3 (x \cdot 2) + 9}{2}$

Step 2.2.9.1.2

Factor $3$ out of $9$ .

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

Step 2.2.9.1.3

Factor $3$ out of $3 (x \cdot 2) + 3 \cdot 3$ .

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

Step 2.2.9.2

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

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

Step 2.2.10

Reorder terms.

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

Step 2.2.11

Remove parentheses.

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

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 $y = m x + b$ .

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

$b = 0$

Step 3.2

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

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

y-intercept: $(0, 0)$

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

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.

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

Step 4.1.2

Multiply $2$ by $3$ .

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

Step 4.1.3

Multiply $3$ by $3$ .

$y = - \frac{6 x + 9}{2}$

Step 4.1.4

Split the fraction $\frac{6 x + 9}{2}$ into two fractions.

$y = - (\frac{6 x}{2} + \frac{9}{2})$

Step 4.1.5

Cancel the common factor of $6$ and $2$ .

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Step 4.1.5.1

Factor $2$ out of $6 x$ .

$y = - (\frac{2 (3 x)}{2} + \frac{9}{2})$

Step 4.1.5.2

Cancel the common factors.

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Step 4.1.5.2.1

Factor $2$ out of $2$ .

$y = - (\frac{2 (3 x)}{2 (1)} + \frac{9}{2})$

Step 4.1.5.2.2

Cancel the common factor.

$y = - (\frac{2 (3 x)}{2 \cdot 1} + \frac{9}{2})$

Step 4.1.5.2.3

Rewrite the expression.

$y = - (\frac{3 x}{1} + \frac{9}{2})$

Step 4.1.5.2.4

Divide $3 x$ by $1$ .

$y = - (3 x + \frac{9}{2})$

Step 4.1.6

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - (3 x \cdot \frac{2}{2} + \frac{9}{2})$

Step 4.1.7

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

$y = - (\frac{3 x \cdot 2}{2} + \frac{9}{2})$

Step 4.1.8

Combine the numerators over the common denominator.

$y = - \frac{3 x \cdot 2 + 9}{2}$

Step 4.1.9

Simplify the numerator.

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Step 4.1.9.1

Factor $3$ out of $3 x \cdot 2 + 9$ .

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Step 4.1.9.1.1

Factor $3$ out of $3 x \cdot 2$ .

$y = - \frac{3 (x \cdot 2) + 9}{2}$

Step 4.1.9.1.2

Factor $3$ out of $9$ .

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

Step 4.1.9.1.3

Factor $3$ out of $3 (x \cdot 2) + 3 \cdot 3$ .

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

Step 4.1.9.2

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

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

Step 4.1.10

Reorder terms.

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

Step 4.1.11

Remove parentheses.

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

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 $x$ .

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

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

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

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Step 4.2.2.3.1.1.1

Apply the distributive property.

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

Step 4.2.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.3.1.1.2.1

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

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

Step 4.2.2.3.1.1.2.2

Factor $2$ out of $2 x$ .

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

Step 4.2.2.3.1.1.2.3

Cancel the common factor.

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

Step 4.2.2.3.1.1.2.4

Rewrite the expression.

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

Step 4.2.2.3.1.1.3

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

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Step 4.2.2.3.1.1.3.1

Multiply $3$ by $- 1$ .

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

Step 4.2.2.3.1.1.3.2

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

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

Step 4.2.2.3.1.1.3.3

Multiply $- 3$ by $3$ .

$- \frac{2}{3} (- 3 x + \frac{- 9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.4

Move the negative in front of the fraction.

$- \frac{2}{3} (- 3 x - \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.5

Apply the distributive property.

$- \frac{2}{3} (- 3 x) - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.6

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.6.1

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

$\frac{- 2}{3} (- 3 x) - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.6.2

Factor $3$ out of $- 3 x$ .

$\frac{- 2}{3} (3 (- x)) - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.6.3

Cancel the common factor.

$\frac{- 2}{3} (3 (- x)) - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.6.4

Rewrite the expression.

$- 2 (- x) - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.7

Multiply $- 1$ by $- 2$ .

$2 x - \frac{2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.8

Cancel the common factor of $2$ .

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Step 4.2.2.3.1.1.8.1

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

$2 x + \frac{- 2}{3} (- \frac{9}{2}) = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.8.2

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$2 x + \frac{- 2}{3} \cdot \frac{- 9}{2} = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.8.3

Factor $2$ out of $- 2$ .

$2 x + \frac{2 (- 1)}{3} \cdot \frac{- 9}{2} = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.8.4

Cancel the common factor.

$2 x + \frac{2 \cdot - 1}{3} \cdot \frac{- 9}{2} = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.8.5

Rewrite the expression.

$2 x + \frac{- 1}{3} \cdot - 9 = - \frac{2}{3} \cdot 0$

Step 4.2.2.3.1.1.9

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.9.1

Factor $3$ out of $- 9$ .

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

Step 4.2.2.3.1.1.9.2

Cancel the common factor.

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

Step 4.2.2.3.1.1.9.3

Rewrite the expression.

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

Step 4.2.2.3.1.1.10

Multiply $- 1$ by $- 3$ .

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

Step 4.2.2.3.2

Simplify the right side.

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Step 4.2.2.3.2.1

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

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Step 4.2.2.3.2.1.1

Multiply $0$ by $- 1$ .

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

Step 4.2.2.3.2.1.2

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

$2 x + 3 = 0$

Step 4.2.2.4

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 4.2.2.5

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 4.2.2.5.1

Divide each term in $2 x = - 3$ by $2$ .

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

Step 4.2.2.5.2

Simplify the left side.

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Step 4.2.2.5.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.5.2.1.1

Cancel the common factor.

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

Step 4.2.2.5.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.2.5.3

Simplify the right side.

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Step 4.2.2.5.3.1

Move the negative in front of the fraction.

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

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3}{2}, 0)$

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 $y$ .

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

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Remove parentheses.

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

Multiply $2$ by $0$ .

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

Step 4.3.2.2.2

Add $0$ and $3$ .

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

Step 4.3.2.2.3

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

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Step 4.3.2.2.3.1

Multiply $3$ by $- 1$ .

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

Step 4.3.2.2.3.2

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

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

Step 4.3.2.2.3.3

Multiply $- 3$ by $3$ .

$y = \frac{- 9}{2}$

Step 4.3.2.2.4

Move the negative in front of the fraction.

$y = - \frac{9}{2}$

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{9}{2})$

Step 4.4

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

$\begin{array}{c} x & y \\ - \frac{3}{2} & 0 \\ 0 & - \frac{9}{2} \end{array}$

Step 5

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

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

y-intercept: $(0, 0)$

$\begin{array}{c} x & y \\ - \frac{3}{2} & 0 \\ 0 & - \frac{9}{2} \end{array}$

Step 6