Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals3h 25m
5. Graphical Applications of Derivatives
Curve Sketching
Problem 33
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = (x2 + 12)/(2x + 1)

1
Identify the domain of the function f(x) = \frac{x^2 + 12}{2x + 1}. The function is undefined where the denominator is zero, so solve 2x + 1 = 0 to find x = -\frac{1}{2}. Thus, the domain is all real numbers except x = -\frac{1}{2}.
Determine the vertical asymptote by setting the denominator equal to zero. Since 2x + 1 = 0 at x = -\frac{1}{2}, there is a vertical asymptote at x = -\frac{1}{2}.
Find the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both the numerator and the denominator are of degree 2 and 1 respectively, divide the leading coefficients: \frac{1}{2}. Therefore, the horizontal asymptote is y = \frac{1}{2}.
Calculate the x-intercepts by setting the numerator equal to zero: x^2 + 12 = 0. Since this equation has no real solutions, there are no x-intercepts.
Evaluate the behavior of the function as x approaches the vertical asymptote and as x approaches infinity to understand the end behavior. This will help in sketching the graph accurately, showing how the function behaves near the asymptotes and at extreme values of x.
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