Math Problem Statement
graph of f(x)=3x+4/2x²+3x+1
Solution
To graph the function , follow these steps:
1. Identify the asymptotes:
-
Vertical Asymptotes: These occur where the denominator is zero, i.e., solve . Using the quadratic formula: So, the roots are and . Hence, there are vertical asymptotes at and .
-
Horizontal Asymptote: Since the degree of the denominator is higher than the degree of the numerator (degree 2 in the denominator and degree 1 in the numerator), the horizontal asymptote is .
2. Find the intercepts:
- x-intercept: Set the numerator , giving . So, the x-intercept is at .
- y-intercept: Set in the function: So, the y-intercept is at .
3. Behavior near asymptotes and end behavior:
- As or , the function approaches the horizontal asymptote .
- Near the vertical asymptotes and , the function will approach infinity or negative infinity, depending on the direction.
4. Plot key points:
- Vertical asymptotes at and .
- Horizontal asymptote at .
- x-intercept at and y-intercept at .
Would you like me to generate a plot of this graph or need more details on any step?
Here are 5 related questions:
- How do you determine slant asymptotes for a rational function?
- What is the significance of the degree of the numerator and denominator in determining asymptotes?
- How do you find the local maxima and minima of rational functions?
- What happens when the degree of the numerator is higher than the denominator?
- How do you analyze the concavity of rational functions?
Tip: Always check both the numerator and denominator when finding intercepts and asymptotes for rational functions.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Intercepts
End Behavior
Formulas
Quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a
Rational Function Asymptotes: Horizontal, Vertical
Theorems
Degree of numerator vs. denominator theorem for asymptotes
Suitable Grade Level
Grades 9-12
Related Recommendation
Solving for Intercepts and Asymptotes of Rational Function f(x) = (3x + 4) / (2x² + 3x + 1)
Graphing Rational Function f(x) = (3x + 3) / (2x - 4)
Graphing Rational Function with Intercepts and Asymptotes: f(x) = (x^2 - 4)/(x^2 - 3x + 2)
Graphing Rational Function f(x) = 3 / (2x + 1)
Graphing Rational Function f(x) = (2x + 2)/x with Asymptotes