Trigonometry Examples

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

Step 1

Find the asymptotes.

Tap for more steps...

Find where the expression $\log_{2} (x - 2) + 4$ is undefined.

$x \leq 2$

Since $\log_{2} (x - 2) + 4$ $\to$ $\infty$ as $x$ $\to$ $2$ from the left and $\log_{2} (x - 2) + 4$ $\to$ $- \infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

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).

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Step 2

Find the point at $x = 3$ .

Tap for more steps...

Replace the variable $x$ with $3$ in the expression.

$f (3) = \log_{2} ((3) - 2) + 4$

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

Subtract $2$ from $3$ .

$f (3) = \log_{2} (1) + 4$

Logarithm base $2$ of $1$ is $0$ .

$f (3) = 0 + 4$

Add $0$ and $4$ .

$f (3) = 4$

The final answer is $4$ .

$4$

Convert $4$ to decimal.

$y = 4$

Step 3

Find the point at $x = 4$ .

Tap for more steps...

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

$f (4) = \log_{2} ((4) - 2) + 4$

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

Subtract $2$ from $4$ .

$f (4) = \log_{2} (2) + 4$

Logarithm base $2$ of $2$ is $1$ .

$f (4) = 1 + 4$

Add $1$ and $4$ .

$f (4) = 5$

The final answer is $5$ .

$5$

Convert $5$ to decimal.

$y = 5$

Step 4

Find the point at $x = 6$ .

Tap for more steps...

Replace the variable $x$ with $6$ in the expression.

$f (6) = \log_{2} ((6) - 2) + 4$

Simplify the result.

Tap for more steps...

Simplify each term.

Tap for more steps...

Subtract $2$ from $6$ .

$f (6) = \log_{2} (4) + 4$

Logarithm base $2$ of $4$ is $2$ .

$f (6) = 2 + 4$

Add $2$ and $4$ .

$f (6) = 6$

The final answer is $6$ .

$6$

Convert $6$ to decimal.

$y = 6$

Step 5

The log function can be graphed using the vertical asymptote at $x = 2$ and the points $(3, 4), (4, 5), (6, 6)$ .

Vertical Asymptote: $x = 2$

$\begin{array}{c} x & y \\ 3 & 4 \\ 4 & 5 \\ 6 & 6 \end{array}$

Step 6