Curve Sketching: Guidelines and Techniques
Welcome to Professor Baker's Math Class! In this lesson, we're diving into the art and science of curve sketching. Understanding how to sketch curves is a fundamental skill in calculus, allowing us to visualize functions and their behavior. These notes provide a checklist intended as a guide to sketching a curve $y = f(x)$ by hand. Remember, not every item is relevant to every function.
A. Domain
First, determine the domain $D$ of the function $f$. The domain is the set of all possible $x$ values for which $f(x)$ is defined. This is a crucial first step because it tells you where the function exists.
B. Intercepts
- The $y$-intercept is $f(0)$. This tells you where the curve intersects the $y$-axis.
- To find the $x$-intercepts, set $y = 0$ and solve for $x$. This tells you where the curve intersects the $x$-axis. Note: this step can be omitted if the equation is difficult to solve.
C. Symmetry
Symmetry can significantly simplify curve sketching.
- Even Function: If $f(-x) = f(x)$ for all $x$ in $D$, then $f$ is an even function, and the curve is symmetric about the $y$-axis.
- Odd Function: If $f(-x) = -f(x)$ for all $x$ in $D$, then $f$ is an odd function, and the curve is symmetric about the origin.
- 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 $p$ is the period.
D. Asymptotes
Asymptotes indicate the behavior of the function as $x$ approaches certain values or infinity.
- Horizontal Asymptotes: If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.
- Vertical Asymptotes: If $\lim_{x \to a^+} f(x) = \pm \infty$ or $\lim_{x \to a^-} f(x) = \pm \infty$, then $x = a$ is a vertical asymptote. These often occur where the denominator of a rational function equals zero.
- Slant Asymptotes: These occur when the degree of the numerator of a rational function is one greater than the degree of the denominator.
E. Intervals of Increase or Decrease
Use the first derivative, $f'(x)$, to determine where the function is increasing or decreasing.
- If $f'(x) > 0$, then $f$ is increasing.
- If $f'(x) < 0$, then $f$ is decreasing.
F. 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 identify local maxima and minima.
G. Concavity and Points of Inflection
Use the second derivative, $f''(x)$, to determine the concavity of the function.
- If $f''(x) > 0$, then $f$ is concave up.
- If $f''(x) < 0$, then $f$ is concave down.
- Inflection points occur where the concavity changes (i.e., where $f''(x) = 0$ or is undefined).
By systematically working through these steps, you can create accurate and informative sketches of various functions. Keep practicing, and you'll become a curve-sketching pro in no time! Good luck, and happy sketching!