Summary Curve Sketching Section 4.5

THINGS TO CONSIDER BEFORE SKETCHING A CURVE Domain Intercepts Symmetry - even, odd, periodic. Asymptotes - vertical, horizontal, slant. Intervals of increase or decrease. Local maximum or minimum values. Concavity and Points of Inflections Not every item above is relevant to every function.

PROCEDURE FOR CURVE SKETCHING Step 1: Pre-Calculus analysis (a) Check the domain of the function to see if any regions of the plane are excluded. (b) Find the x- and y-intercepts. (c) Test for symmetry with respect to the y-axis and the origin. (Is the function even or odd?)

PROCEDURE (CONTINUED) Step 2: Calculus Analysis (a) Find the asymptotes (vertical, horizontal, and/or slant). (b) Use the first derivative to find the critical points and to find the intervals where the graph is increasing and decreasing. (c) Test the critical points for local maxima and local minima. (d) Use the second derivative to find the intervals where the graph is concave up and concave down and to locate inflection points.

PROCEDURE (CONCLUDED) Step 3: Plot a few points (including all critical points, inflection points, and intercepts). Step 4: Sketch the graph. (NOTE: On the graph, label all critical points, inflection points, intercepts and asymptotes.)

Curve Sketching Guidelines for sketching a curve: Domain Intercepts Determine D, the set of values of x for which f (x) is defined Intercepts The y-intercept is f(0) To find the x-intercept, set y=0 and solve for x Symmetry 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 If f (-x) = - f (x) for all x in D, then f is an odd function and the curve is symmetric about the origin Asymptotes Horizontal asymptotes Vertical asymptotes

Guidelines for sketching a curve (cont.): E. Intervals of Increase or Decrease f is increasing where f ′ (x) > 0 f is decreasing where f ′ (x) < 0 F. Local Maximum and Minimum Values Find the critical numbers of f ( f ′ (c)=0 or f ′ (c) doesn’t exist) If f ′ is changing from positive to negative at a critical number c, then f (c) is a local maximum If f ′ is changing from negative to positive at a critical number c, then f (c) is a local minimum G. Concavity and Inflection Points f is concave upward where f ′ ′ (x) > 0 f is concave downward where f ′ ′ (x) < 0 Inflection points occur where the direction of concavity changes H. Sketch the Curve

DISAIMIS O M A I N N T E R C E P T S Y M M E T R Y S Y M P T O T E S N T E R V A L S A X M I N N F L E C T I O N K E T C H

New Information You know about vertical asymptotes You know about horizontal asymptotes What about other asymptotes?

Example

SLANT ASYMPTOTES The line y = mx + b is a slant asymptote of the graph of y = f (x) if

SLANT ASYMPTOTES AND RATIONAL FUNCTIONS For rational functions, slant asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. For rational functions, the slant asymptote can be found by using long division of polynomials.

Example: D

Example: I

Example: S

Example: A

Example: A

Example: A

Example: I

Example: M

Example: I

Sketch