2. CURVE SKETCHING Emily Cooper and Ashli Haas © Emily Cooper and Ashli Haas 2011

3. INTRODUCTION  The purpose of curve sketching is to graph the curve of a function by using differentiation.

4. HOW TO SKETCH THE CURVE  Step 1 – Find critical points (x-intercepts)  Step 2 – Fine horizontal, vertical or oblique asymptotes  Step 3 – Take the derivative of the original equation  Step 4 – Sign line  Step 5 – Determine where the graph is increasing and decreasing  Step 6 - Graph

5. FINDING X-INTERCEPTS

7. ASYMPTOTES  An asymptote to a curve is a straight line to which the curve approaches, but never touches, as the distance from the origin increases.  There are three types of possible asymptotes: Vertical, Horizontal and Oblique.

8. VERTICAL ASYMPTOTES

9.  Vertical asymptotes are x-values and are not able to be crossed.

10. HORIZONTAL ASYMPTOTES  To calculate the horizontal asymptotes, take the ratio of the coefficients of the variable raised to the highest power in the given problem.

11. HORIZONTAL ASYMPTOTES  If the powers of the variables are equal, the h.a. is the ratio of the coefficients.  If the power of the variable in the denominator is larger, then y=0  If the power of the variable in the numerator is larger, there is an oblique asymptote

12. HORIZONTAL ASYMPTOTES

13.  Horizontal asymptotes are y-values and can be crossed no more than once.

14. OBLIQUE ASYMPTOTES

15.  Oblique asymptotes are neither horizontal nor vertical but diagonal instead.

16. OBLIQUE ASYMPTOTES  Oblique asymptotes cannot be crossed.

17. DERIVATIVE  Take the derivative of the original equation.  To take a derivative, you multiply each coefficient by the corresponding variable’s exponent then lower the exponent by 1.  For example:f(x) = 20x 3 – 3x 5 f’(x) = 60x 2 – 15x 4 Simplify the derivative f’(x) = 15x 2 (4-x 2 ) f’(x) = (3x)(5x)(2-x)(2+x)

18. DERIVATIVE  To take the derivative of a fraction, you must take the denominator and multiply it by the derivative of the numerator. Then subtract from that the numerator multiplied by the derivative of the denominator. Then divide by the denominator squared.

19. DERIVATIVE

20. DERIVATIVE & SIGN LINE  Using the derivative, you must find critical points for your sign line.  Factor your derivative to find the portions on the side of the sign line. Then solve for x in each of those to find points on your sign line.

21. DERIVATIVE & SIGN LINE

22. SIGN LINE  Sign lines are meant to tell where the graph of the equation is increasing or decreasing.  Use the portions of the derivative to do the sign line.  To find critical points on the sign line, solve for x for each of the portions.  Those are the points listed on the top of your line.

23. SIGN LINE  When the graph of a portion is positive, it is shown as a solid line.  When the graph of portion becomes negative, it is shown as a dashed line  Determine whether the graph is negative or positive by using your critical points

24. SIGN LINE  The points that are decreasing will have a negative slope between the given intervals and will follow the asymptotes downwards.  The points that are increasing will have a positive slope between the given intervals and will follow the asymptote upwards.

25. GRAPH THE CURVE

27. EXAMPLE 1

28. EXAMPLE 2

29. TRY ME!

30. SOLUTION #1

31. SOLUTION #2

32. SOLUTION #3