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Section 3.6 – Curve Sketching
Guidelines for sketching a CurveThe following checklist is intended as a guide to sketching
a curve by hand without a calculator. Not every item is relevant to every function. But the guidelines provide all the information needed to make a sketch that displays the most important aspects of the function.
1. Domain
2. x and y-intercept(s)
3. Symmetry
4. Horizontal, Vertical, and Slant Asymptotes
5. Intervals of Increase or Decrease
6. Local Maximum and Minimum Values
7. Concavity and Points of Inflection
Once you have completed the checklist, sketch the graph.
Example 1Without a calculator, sketch the graph of the function below
by finding everything on the checklist (if it exists).
1. Domain
2. x and y-intercept(s)
3. Symmetry
4. Horizontal, Vertical, and Slant Asymptotes
5. Intervals of Increase or Decrease
6. Local Maximum and Minimum Values
7. Concavity and Points of Inflection
2
221xx
f x
Mr. Wells will approximate
irrational numbers for
you.
Answer to Example 1
2
221xx
f x
Even Function.
Example 2Without a calculator, sketch the graph of the function below
by finding everything on the checklist (if it exists).
1. Domain
2. x and y-intercept(s)
3. Symmetry
4. Horizontal, Vertical, and Slant Asymptotes
5. Intervals of Increase or Decrease
6. Local Maximum and Minimum Values
7. Concavity and Points of Inflection
4 34f x x x Mr. Wells will approximate
irrational numbers for
you.
Answer to Example 2
4 34f x x x Inflection
Points
Example 3Without a calculator, sketch the graph of the function below
by finding everything on the checklist (if it exists).
1. Domain
2. x and y-intercept(s)
3. Symmetry
4. Horizontal, Vertical, and Slant Asymptotes
5. Intervals of Increase or Decrease
6. Local Maximum and Minimum Values
7. Concavity and Points of Inflection
3
2 1xx
f x
Mr. Wells will approximate
irrational numbers for
you.
Answer to Example 3
3
2 1xx
f x
3 343,
3 343,
y = x
Odd Function.
Example 4Without a calculator, sketch the graph of the function below
by finding everything on the checklist (if it exists).
1. Domain
2. x and y-intercept(s)
3. Symmetry
4. Horizontal, Vertical, and Slant Asymptotes
5. Intervals of Increase or Decrease
6. Local Maximum and Minimum Values
7. Concavity and Points of Inflection
sin cos 0,2f x x x on Mr. Wells will approximate
irrational numbers for
you.
Answer to Example 4
sin cos
0,2
f x x x
on
Inflection Points
2
4 , 2
54 , 2
Without a calculator, sketch the graph of the function below by finding everything on the checklist (if it exists).
1.Domain2.x and y-intercept(s)3.Symmetry4.Horizontal, Vertical, and Slant Asymptotes5.Intervals of Increase or Decrease6.Local Maximum and Minimum Values7.Concavity and Points of Inflection
Example 5
2 4xxf x
Answer to Example 5
2 4xxf x
2,4
2, 4
y = xx = 0
Odd Function.
