Math Problem Statement
Graph the following function. Determine the asymptotes, if any. List the domain and the x-intercepts and the y-intercepts. Check your work using a graphing calculator. f(x)= x^2-16 / x+1
Solution
To analyze the function , we will determine its asymptotes, domain, x-intercepts, and y-intercepts.
1. Asymptotes
Vertical Asymptote:
- Vertical asymptotes occur where the denominator is zero, and the numerator is not zero.
- Setting the denominator equal to zero:
- Thus, there is a vertical asymptote at .
Horizontal Asymptote:
- For rational functions, the horizontal asymptote can be found by comparing the degrees of the numerator and the denominator.
- The degree of the numerator is 2, and the degree of the denominator is 1.
- Since the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote. However, there is an oblique asymptote, which can be found through polynomial long division.
Polynomial Long Division: This implies: Thus, the oblique asymptote is .
2. Domain
- The domain of includes all real numbers except where the denominator is zero: Thus, the domain is .
3. Intercepts
X-Intercepts:
- To find the x-intercepts, set : Factoring gives: So, the x-intercepts are and .
Y-Intercept:
- To find the y-intercept, evaluate : So, the y-intercept is .
Summary
- Vertical Asymptote:
- Oblique Asymptote:
- Domain:
- X-Intercepts: and
- Y-Intercept:
Graphing Check
You can verify these findings using a graphing calculator by plotting the function .
If you have any questions or would like further details on any of the steps, let me know!
Further Exploration
- How do changes in the coefficients of the function affect the asymptotes?
- What would happen to the function if the denominator were quadratic instead of linear?
- Can you find the limits of the function as approaches the asymptotes?
- How do the intercepts relate to the behavior of the graph?
- What are the characteristics of rational functions that can help predict their graph?
Tip: When analyzing rational functions, always check for asymptotes first to understand the overall shape of the graph.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Intercepts
Domain
Formulas
Vertical Asymptote: x + 1 = 0
Oblique Asymptote: y = x - 1
X-Intercepts: f(x) = 0
Y-Intercept: f(0)
Theorems
Properties of Asymptotes for Rational Functions
Intercepts of Functions
Suitable Grade Level
Grades 9-11
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