CURVE SKETCHINGLesson 5.4

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

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Tools for Curve Sketching

Test for concavity

Test for increasing/decreasing functions

Critical points

Zeros

Maximums and Minimums

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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

Using First, Second Derivatives

Note the four possibilities for a function to be … • Increasing or decreasing• Concave up or concave down

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Positive(increasing function)

Negative (decreasing

function)

Positive (concave up)

Negative (concave

down)

f '(x)

f ''(x)

Try It Out

Find as much as you can about the function without graphing it on the calculator

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3 215( ) 18 1

2f x x x x= − − −

2 1

xy

x=

−( ) lnf x x x= −

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)

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'( ) 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