Calculus Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{x - 2} - 3 = 0$ .

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

Step 1.2.2

Add $3$ to both sides of the equation.

$\frac{1}{x - 2} = 3$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1$

Step 1.2.3.2

Remove parentheses.

$x - 2, 1$

Step 1.2.3.3

The LCM of one and any expression is the expression.

$x - 2$

Step 1.2.4

Multiply each term in $\frac{1}{x - 2} = 3$ by $x - 2$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $\frac{1}{x - 2} = 3$ by $x - 2$ .

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $x - 2$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.1.2

Rewrite the expression.

$1 = 3 (x - 2)$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Apply the distributive property.

$1 = 3 x + 3 \cdot - 2$

Step 1.2.4.3.2

Multiply $3$ by $- 2$ .

$1 = 3 x - 6$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Rewrite the equation as $3 x - 6 = 1$ .

$3 x - 6 = 1$

Step 1.2.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.2.1

Add $6$ to both sides of the equation.

$3 x = 1 + 6$

Step 1.2.5.2.2

Add $1$ and $6$ .

$3 x = 7$

Step 1.2.5.3

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 1.2.5.3.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 1.2.5.3.2

Simplify the left side.

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Step 1.2.5.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.5.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 1.2.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{7}{3}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{0 - 2} - 3$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $\frac{1}{(0) - 2} - 3$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $2$ from $0$ .

$y = \frac{1}{- 2} - 3$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{2} - 3$

Step 2.2.3.2

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

$y = - \frac{1}{2} - 3 \cdot \frac{2}{2}$

Step 2.2.3.3

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

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

Step 2.2.3.4

Combine the numerators over the common denominator.

$y = \frac{- 1 - 3 \cdot 2}{2}$

Step 2.2.3.5

Simplify the numerator.

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Step 2.2.3.5.1

Multiply $- 3$ by $2$ .

$y = \frac{- 1 - 6}{2}$

Step 2.2.3.5.2

Subtract $6$ from $- 1$ .

$y = \frac{- 7}{2}$

Step 2.2.3.6

Move the negative in front of the fraction.

$y = - \frac{7}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{7}{2})$

Step 3

List the intersections.

x-intercept(s): $(\frac{7}{3}, 0)$

y-intercept(s): $(0, - \frac{7}{2})$

Step 4