This document provides guidance on curve sketching functions by hand without using a graphing calculator. It outlines the key steps and tools needed for curve sketching, including determining the domain, finding intercepts and critical points, checking for asymptotes, analyzing the first and second derivatives to determine concavity and monotonicity, and connecting the points to sketch the curve. The document also provides examples to demonstrate how to apply these techniques to sketch curves based on information about the function's behavior.

1. CURVE SKETCHING
Lesson 5.4

2. Motivation
Graphing calculators decrease the
importance of curve sketching
So why a lesson on curve sketching?
A calculator graph may be misleading
• What happens outside specified window?
• Calculator plots, connects points without
showing what happens between points
• False asymptotes
Curve sketching is a good way to reinforce
concepts of lessons in this chapter
2

3. Tools for Curve Sketching
Test for concavity
Test for increasing/decreasing functions
Critical points
Zeros
Maximums and Minimums
3

4. Strategy
Determine domain of function
Find y-intercepts, x-intercepts (zeros)
Check for vertical, horizontal asymptotes
Determine values for f '(x) = 0, critical points
Determine f ''(x)
• Gives inflection points
• Test for intervals of concave up, down
Plot intercepts, critical points, inflection points
Connect points with smooth curve
Check sketch with graphing calculator 4

5. Using First, Second Derivatives
Note the four possibilities for a function to
be …
• Increasing or decreasing
• Concave up or concave down
5
Positive
(increasing
function)
Negative
(decreasing
function)
Positive
(concave up)
Negative
(concave
down)
f '(x)
f ''(x)

6. Try It Out
Find as much as you can about the
function without graphing it on the
calculator
6
3 215
( ) 18 1
2
f x x x x= − − −
2
1
x
y
x
=
−
( ) lnf x x x= −

7. Graphing Without the Formula
Consider a function of this description
• Can you graph it?
This function is continuous for all reals
•
•
•
•
• A y-intercept at (0, 2)
7
'( ) 0 on (- , -6) and (1, 3)f x < ∞
'( ) 0 on (-6, 1) and (3, )f x > ∞
''( ) 0 on (- , -6) and (3, )f x > ∞ ∞
''( ) < 0 on interval (-6, 3)f x