L5 Curve Sketching - Part 1
Completion requirements
Unit 3
Curve Sketching
Lesson 5: Curve Sketching
Asymptotes, symmetry, extrema, and concavity! In the previous four Lessons, the necessary curve sketching skills were introduced. In this Lesson, all of those skills will be used in combination with the many skills learned in previous
math courses to sketch complicated functions without technology.
Steps to Successful Curve Sketching
Although not all of the following steps will be followed to sketch every curve, they serve as useful reminders of which features warrant attention.
Steps to Successful Curve Sketching
Although not all of the following steps will be followed to sketch every curve, they serve as useful reminders of which features warrant attention.
Step 1:
Determine the domain of the function.
Step 2:
Find the intercepts, if possible, to see if and where the graph crosses the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»- and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axes.
Step 3:
Locate the asymptotes – vertical, horizontal, oblique. In doing so, the behaviour of the function at any undefined values or points of discontinuity will become known as will the behaviour of the function as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8594;«/mo»«mo»§#177;«/mo»«mo»§#8734;«/mo»«/mrow»«/mstyle»«/math».
Step 4:
Determine the symmetry of the function. Is it odd, even, or neither?
Step 5:
Find the first derivative to determine the critical points and the intervals of increase and decrease.
Step 6:
Locate the local extrema. Calculate the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates at these critical points. The intervals of increase and
decrease or the Second Derivative Test can be used to find the local extrema.
Step 7:
Find the second derivative to determine the concavity and the inflection point(s).
Step 8:
Sketch a smooth curve through the points determined above. Be mindful of any discontinuities in the graph of the function.