Fall 2025: Chapter 4.5 and 4.6 - Curve Sketching

Welcome back to Professor Baker's Math Class! In this post, we'll be exploring the concepts covered in Chapters 4.5 and 4.6, focusing on curve sketching. This is where we bring together many of the tools we've learned to visualize and understand functions.

Key Concepts for Curve Sketching

Here's a breakdown of the essential elements we'll use to create accurate curve sketches:

  1. Domain: It's always a great starting point! The domain, denoted as $D$, of a function $f(x)$ is the set of all possible $x$ values for which $f(x)$ is defined. Understanding the domain helps us identify where the function exists.
  2. Intercepts:
    • Y-intercept: This is the point where the curve intersects the y-axis. It's found by evaluating $f(0)$.
    • X-intercept(s): These are the points where the curve intersects the x-axis. To find them, we set $y = 0$ and solve for $x$. Sometimes, this step can be omitted if it's too difficult to solve.
  3. Symmetry: Symmetry can significantly simplify the sketching process.
    • Even Function: If $f(-x) = f(x)$ for all $x$ in the domain $D$, then $f$ is an even function, and the curve is symmetric about the y-axis. Examples include $y = x^2$, $y = x^4$, $y = |x|$, and $y = \cos(x)$.
    • Odd Function: If $f(-x) = -f(x)$ for all $x$ in the domain $D$, then $f$ is an odd function, and the curve is symmetric about the origin. Examples include $y = x$, $y = x^3$, $y = 1/x$, and $y = \sin(x)$.
    • Periodic Function: If $f(x + p) = f(x)$ for all $x$ in $D$, where $p$ is a positive constant, then $f$ is a periodic function, and the smallest such number $p$ is called the period. For example, $y = \sin(x)$ has a period of $2\pi$, and $y = \tan(x)$ has a period of $\pi$.
  4. Asymptotes: Asymptotes define the behavior of the function as it approaches certain values or infinity.
    • Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then the line $y = L$ is a horizontal asymptote.
    • Vertical Asymptotes: The line $x = a$ is a vertical asymptote if at least one of the following is true: $$ \lim_{x \to a^+} f(x) = \infty, \quad \lim_{x \to a^-} f(x) = \infty, \quad \lim_{x \to a^+} f(x) = -\infty, \quad \lim_{x \to a^-} f(x) = -\infty $$ For rational functions, you can usually find vertical asymptotes by setting the denominator equal to zero (after simplifying).
    • Slant Asymptotes: These are more advanced and will be discussed later.
  5. Intervals of Increase or Decrease: Use the I/D Test! Compute $f'(x)$ and find the intervals where $f'(x) > 0$ (function is increasing) and where $f'(x) < 0$ (function is decreasing).
  6. Local Maximum or Minimum Values: Find the critical numbers of $f$ (where $f'(x) = 0$ or $f'(x)$ does not exist). Use the First or Second Derivative Test to determine if these points are local maxima or minima.
  7. Concavity and Points of Inflection: Compute $f''(x)$ and use the Concavity Test. The curve is concave upward where $f''(x) > 0$ and concave downward where $f''(x) < 0$. Inflection points occur where the concavity changes.

Sketching the Curve

Finally, combine all the information above to sketch the curve! Sketch asymptotes as dashed lines. Plot intercepts, maximum and minimum points, and inflection points. Then, draw the curve passing through these points, rising and falling according to your analysis of $f'(x)$, with concavity determined by $f''(x)$, and approaching the asymptotes.

Remember, practice makes perfect! Work through plenty of examples, and don't be afraid to make mistakes. Each sketch is a learning opportunity. You've got this!