Detailed curve sketching examples: applying asymptotes and end behavior.

📚 Quick Study Guide

• 🔍 Asymptotes: These are lines that the curve approaches but never touches. There are three types: vertical, horizontal, and oblique.
• 📏 Vertical Asymptotes: Occur where the function is undefined (usually where the denominator of a rational function equals zero). To find them, set the denominator equal to zero and solve for $x$.
• ↔️ Horizontal Asymptotes: Determined by examining the limit of the function as $x$ approaches positive and negative infinity. If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.
• ↗️ Oblique (Slant) Asymptotes: Occur when the degree of the numerator is exactly one greater than the degree of the denominator. Find them by performing polynomial long division. The quotient (ignoring the remainder) is the equation of the oblique asymptote.
• 🧭 End Behavior: Describes what happens to the function as $x$ approaches positive and negative infinity. This is closely related to horizontal and oblique asymptotes.
• 📈 Curve Sketching Steps:
  1. Find the domain of the function.
  2. Find the intercepts (where the graph crosses the x and y axes).
  3. Find the asymptotes (vertical, horizontal, and oblique).
  4. Determine the end behavior.
  5. Find critical points (where the derivative is zero or undefined).
  6. Find inflection points (where the second derivative is zero or undefined).
  7. Analyze the sign of the first and second derivatives to determine increasing/decreasing intervals and concavity.
  8. Sketch the graph!

🧪 Practice Quiz

1. Question 1: What is the vertical asymptote of the function $f(x) = \frac{1}{x-2}$?
1. A: $x = 0$
2. B: $x = 2$
3. C: $y = 0$
4. D: $y = 2$
2. Question 2: Which of the following functions has a horizontal asymptote at $y = 0$?
1. A: $f(x) = \frac{x}{x^2 + 1}$
2. B: $f(x) = \frac{x^2}{x + 1}$
3. C: $f(x) = x$
4. D: $f(x) = \frac{x^2 + 1}{x}$
3. Question 3: Which of the following indicates the presence of an oblique asymptote?
1. A: Degree of numerator equals the degree of the denominator.
2. B: Degree of numerator is less than the degree of the denominator.
3. C: Degree of numerator is one greater than the degree of the denominator.
4. D: Degree of numerator is two greater than the degree of the denominator.
4. Question 4: What is the end behavior of the function $f(x) = x^3$ as $x$ approaches infinity?
1. A: $f(x)$ approaches 0
2. B: $f(x)$ approaches $-\infty$
3. C: $f(x)$ approaches $\infty$
4. D: $f(x)$ approaches 1
5. Question 5: What is the horizontal asymptote of the function $f(x) = \frac{3x^2}{x^2 + 1}$?
1. A: $y = 0$
2. B: $y = 1$
3. C: $y = 3$
4. D: There is no horizontal asymptote.
6. Question 6: Which function has a vertical asymptote at x = -1 and a horizontal asymptote at y = 2?
1. A: $f(x) = \frac{2x}{x+1}$
2. B: $f(x) = \frac{2x+2}{x+1}$
3. C: $f(x) = \frac{2x^2}{x+1}$
4. D: $f(x) = \frac{2x+5}{x+1}$
7. Question 7: What is the oblique asymptote of $f(x) = \frac{x^2 + 1}{x}$?
1. A: $y = x$
2. B: $y = 1$
3. C: $y = x + 1$
4. D: There is no oblique asymptote.