Precalculus Examples

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

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{x + 2}{x - 6}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x + 2 = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{(0) + 2}{(0) - 6}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0 + 2}{(0) - 6}$

Step 2.2.2

Remove parentheses.

$y = \frac{0 + 2}{0 - 6}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0) + 2}{(0) - 6}$

Step 2.2.4

Simplify $\frac{(0) + 2}{(0) - 6}$ .

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Step 2.2.4.1

Cancel the common factor of $(0) + 2$ and $(0) - 6$ .

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Step 2.2.4.1.1

Factor $2$ out of $0$ .

$y = \frac{2 (0) + 2}{(0) - 6}$

Step 2.2.4.1.2

Factor $2$ out of $2$ .

$y = \frac{2 \cdot 0 + 2 \cdot 1}{(0) - 6}$

Step 2.2.4.1.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot 1$ .

$y = \frac{2 \cdot (0 + 1)}{(0) - 6}$

Step 2.2.4.1.4

Cancel the common factors.

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Step 2.2.4.1.4.1

Factor $2$ out of $0$ .

$y = \frac{2 \cdot (0 + 1)}{2 (0) - 6}$

Step 2.2.4.1.4.2

Factor $2$ out of $- 6$ .

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

Step 2.2.4.1.4.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 3$ .

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

Step 2.2.4.1.4.4

Cancel the common factor.

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

Step 2.2.4.1.4.5

Rewrite the expression.

$y = \frac{0 + 1}{0 - 3}$

Step 2.2.4.2

Simplify the expression.

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Step 2.2.4.2.1

Add $0$ and $1$ .

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

Step 2.2.4.2.2

Subtract $3$ from $0$ .

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

Step 2.2.4.2.3

Move the negative in front of the fraction.

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{3})$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0)$

y-intercept(s): $(0, - \frac{1}{3})$

Step 4