Trigonometry Examples

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

Find the asymptotes.

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Set the argument of the logarithm equal to zero.

$x^{2} - 4 x = 0$

Solve for $x$ .

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Factor $x$ out of $x^{2} - 4 x$ .

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Factor $x$ out of $x^{2}$ .

$x \cdot x - 4 x = 0$

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

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Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

The vertical asymptote occurs at $x = 0, x = 4$ .

Vertical Asymptote: $x = 0, x = 4$

Find the point at $x = - 1$ .

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Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \log_{3} ({(- 1)}^{2} - 4 \cdot - 1)$

Simplify the result.

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Simplify each term.

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Raise $- 1$ to the power of $2$ .

$f (- 1) = \log_{3} (1 - 4 \cdot - 1)$

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

$f (- 1) = \log_{3} (5)$

The final answer is $\log_{3} (5)$ .

$\log_{3} (5)$

Convert $\log_{3} (5)$ to decimal.

$= 1.46497352$

Find the point at $x = - 2$ .

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Replace the variable $x$ with $- 2$ in the expression.

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

Simplify the result.

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Simplify each term.

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Raise $- 2$ to the power of $2$ .

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

Multiply $- 4$ by $- 2$ .

$f (- 2) = \log_{3} (4 + 8)$

Add $4$ and $8$ .

$f (- 2) = \log_{3} (12)$

The final answer is $\log_{3} (12)$ .

$\log_{3} (12)$

Convert $\log_{3} (12)$ to decimal.

$= 2.2618595$

Find the point at $x = - 3$ .

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

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

Simplify the result.

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Simplify each term.

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Raise $- 3$ to the power of $2$ .

$f (- 3) = \log_{3} (9 - 4 \cdot - 3)$

Multiply $- 4$ by $- 3$ .

$f (- 3) = \log_{3} (9 + 12)$

Add $9$ and $12$ .

$f (- 3) = \log_{3} (21)$

The final answer is $\log_{3} (21)$ .

$\log_{3} (21)$

Convert $\log_{3} (21)$ to decimal.

$= 2.77124374$

The log function can be graphed using the vertical asymptote at $x = 0, x = 4$ and the points $(- 1, 1.46497352), (- 2, 2.2618595), (- 3, 2.77124374)$ .

Vertical Asymptote: $x = 0, x = 4$

$\begin{array}{c} x & y \\ - 3 & 2.771 \\ - 2 & 2.262 \\ - 1 & 1.465 \end{array}$