Section 3-6 Curve Sketching
Steps to Analyze a Graph: Intercepts and symmetry Asymptotes maximums and minimums Increasing & decreasing Points of inflection Concavity graph
Intercepts Intercepts: x-intercept: when y = 0 y-intercept: when x =0
Symmetry About the y-axis: About the origin: About the x-axis: Replace every x with –x if the function is Symmetric about the y-axis (all exponents are even) About the origin: Replace every x with –x if the function is symmetric about the origin (all exponents are odd) About the x-axis: not a function
Asymptotes Only occur in rational functions Vertical: set denominator equal to zero Horizontal: take the limit as x approaches infinity Slant: occur when the degree in the numerator is one higher than the denominator Use long division Rewrite function as y = mx + b + remainder Remainder tends to zero as x approaches infinity, the line y = mx + b is the asymptote
Horizontal Asymptotes BOBO BOTN EATS DC Bigger on bottom: y = 0 Bigger on top: none Exponents are the same: divide coefficients
Maximums and Minimums Use the first derivative test to find the x values Substitute x into the original equation to obtain y-coordinate Points: ordered pair (x,y)
Increasing and Decreasing Find critical points First derivative test Positive—increasing Negative—decreasing increasing increasing
Inflection Points Set 2nd Derivative equal to zero and solve Test for changes in concavity
Concavity 2nd derivative test Positive – concave up Negative- concave down
1) Sketch the curve which has the following: relative max relative min increasing on and decreasing on concave up concave down point of inflection
2.) Sketch the graph of no calculator! a) Intercepts and symmetry b) Asymptotes
2.) Sketch the graph of c) maximums and minimums
2.) Sketch the graph of d) Increasing & decreasing
2.) Sketch the graph of Points of inflection f) Concavity
2.) Sketch the graph of g) Graph
3.) Sketch the graph of no calculator! a) Intercepts b) Asymptotes
3.) Sketch the graph of c) maximums and minimums
3.) Sketch the graph of d) Increasing & decreasing
3.) Sketch the graph of e) Inflection Points
3.) Sketch the graph of f) Concavity
3.) Sketch the graph of g) Graph