1. Chapter 4 – Applications of Differentiation
4.5 Summary of Curve Sketching
4.6 Graphing with Calculus and Calculators
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

2. Guidelines for Sketching a Curve
The following checklist is intended as a guide to sketching a curve y = f (x) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.)
But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function.
4.5 Summary of Curve Sketching Graphing with Calculators and Calculators

3. Guidelines for Sketching a Curve
Domain – Start by determining the domain of f.
Intercepts – The y-intercept is f(0). To find the x-intercept, set y = 0 and solve for x (We omit this if the equation is difficult to solve.)
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

4. Guidelines for Sketching a Curve
Symmetry
If f(-x) = f(x) then f is an even function and symmetric about the y-axis.
If f(-x) = -f(x) then f is an odd function and symmetric about the x-axis.
If f(x+p)=f(x) then f is a periodic function and the pattern of the first period repeats.
Even function: reflection symmetry
Odd function: rotational symmetry
Periodic function: translational symmetry
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

5. Guidelines for Sketching a Curve
Asymptotes
Horizontal Asymptotes.
If then y=L is a horizontal asymptote.
Vertical Asymptotes.
The line x = a is a HA if one of the following statements is true:
Slant Asymptotes.
If then the line y=mx+b is the slant asymptote and it is found using long division.
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

6. Guidelines for Sketching a Curve
Intervals of Increase or Decrease – Use the I/D test to determine where the function is increasing and/or decreasing.
Local Max and Min Values – Find the critical numbers of f then use the First Derivative test.
Concavity and Points of Inflection – Compute the Second Derivative and use the Concavity Test.
Sketch the curve – Use the information from A-G to sketch the curve.
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

7. Example 1
Use the guidelines to sketch the curve of the following functions:
4.5 Summary of Curve Sketching
4.6 Graphing with Calculators and Calculators

8. Example 2
Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use the graphs of f’ and f” to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. .
4.5 Summary of Curve Sketching Graphing with Calculators and Calculators

9. Example 3
Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum, minimum, and inflections points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes. .
4.5 Summary of Curve Sketching Graphing with Calculators and Calculators