Trigonometry Examples

$h (x) \log_{2} (x - 2) + 1$

Step 1

Find the asymptotes.

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Step 1.1

Simplify.

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Step 1.1.1

Subtract $1$ from both sides of the equation.

$y x \log_{2} (x - 2) = - 1$

Step 1.1.2

Divide each term in $y x \log_{2} (x - 2) = - 1$ by $x \log_{2} (x - 2)$ and simplify.

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Step 1.1.2.1

Divide each term in $y x \log_{2} (x - 2) = - 1$ by $x \log_{2} (x - 2)$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

Step 1.1.2.2.2

Cancel the common factor of $\log_{2} (x - 2)$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

Divide $y$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

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

Step 1.2

Find where the expression $- \frac{1}{x \log_{2} (x - 2)}$ is undefined.

$x \leq 2, x = 3$

Step 1.3

Since $- \frac{1}{x \log_{2} (x - 2)}$ $\to$ $\infty$ as $x$ $\to$ $3$ from the left and $- \frac{1}{x \log_{2} (x - 2)}$ $\to$ $- \infty$ as $x$ $\to$ $3$ from the right, then $x = 3$ is a vertical asymptote.

$x = 3$

Step 1.4

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.5

Find $n$ and $m$ .

$n = 0$

$m = 1$

Step 1.6

Since $n < m$ , the x-axis, $y = 0$ , is the horizontal asymptote.

$y = 0$

Step 1.7

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.8

This is the set of all asymptotes.

Vertical Asymptotes: $x = 3$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = 3$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = 4$ .

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Step 2.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 0$

Step 2.2

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

The log function can be graphed using the vertical asymptote at $x = 3$ and the points $(4, 0), (5, 0), (6, 0)$ .

Vertical Asymptote: $x = 3$

$\begin{array}{c} x & y \\ 4 & 0 \\ 5 & 0 \\ 6 & 0 \end{array}$

Step 4