Math Problem Statement
Solution
The function given is:
This is a logarithmic function with a vertical shift. Let's analyze it step-by-step.
1. Understanding the Function
The base of the logarithm is 3, so the function has a basic shape of a logarithmic curve, which generally starts from (approaching a vertical asymptote there) and increases slowly as grows. The "+ 2" means the entire graph of is shifted up by 2 units.
2. Vertical Asymptote
For the function , the vertical asymptote is at , since logarithmic functions are undefined for . This asymptote will be a vertical line at .
3. Plotting Points
Let's find two points on the graph by substituting values for :
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For : So, one point is .
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For : So, another point is .
Summary of Key Information
- Vertical Asymptote:
- Points on the Graph: and
Would you like a graph of this function or more details on any part of the solution?
Related Questions
- What is the effect of changing the base of the logarithm on the graph?
- How does the function change if we add a coefficient to ?
- What is the domain and range of ?
- How would we find the x-intercept of ?
- What happens to the graph if we replace with for some constant ?
Tip
For logarithmic functions of the form , the constant shifts the graph vertically, and the base affects the rate of growth.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Transformations of Functions
Vertical Shifts
Vertical Asymptotes
Formulas
f(x) = c + \log_b{x}
Theorems
Properties of Logarithmic Functions
Suitable Grade Level
Grades 10-12
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