Precalculus Examples

Graph f(x)=- log base 3 of x-1+3
Step 1
Find the asymptotes.
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Step 1.1
Find where the expression is undefined.
Step 1.2
Since as from the left and as from the right, then is a vertical asymptote.
Step 1.3
Ignoring the logarithm, consider the rational function where is the degree of the numerator and is the degree of the denominator.
1. If , then the x-axis, , is the horizontal asymptote.
2. If , then the horizontal asymptote is the line .
3. If , then there is no horizontal asymptote (there is an oblique asymptote).
Step 1.4
There are no horizontal asymptotes because is .
No Horizontal Asymptotes
Step 1.5
No oblique asymptotes are present for logarithmic and trigonometric functions.
No Oblique Asymptotes
Step 1.6
This is the set of all asymptotes.
Vertical Asymptotes:
No Horizontal Asymptotes
Vertical Asymptotes:
No Horizontal Asymptotes
Step 2
Find the point at .
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Step 2.1
Replace the variable with in the expression.
Step 2.2
Simplify the result.
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Step 2.2.1
Simplify each term.
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Step 2.2.1.1
Subtract from .
Step 2.2.1.2
Logarithm base of is .
Step 2.2.1.3
Multiply by .
Step 2.2.2
Add and .
Step 2.2.3
The final answer is .
Step 2.3
Convert to decimal.
Step 3
Find the point at .
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Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Subtract from .
Step 3.2.1.2
Logarithm base of is .
Step 3.2.1.3
Multiply by .
Step 3.2.2
Add and .
Step 3.2.3
The final answer is .
Step 3.3
Convert to decimal.
Step 4
Find the point at .
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Simplify each term.
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Step 4.2.1.1
Subtract from .
Step 4.2.1.2
Logarithm base of is .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Add and .
Step 4.2.3
The final answer is .
Step 4.3
Convert to decimal.
Step 5
The log function can be graphed using the vertical asymptote at and the points .
Vertical Asymptote:
Step 6
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