1. C URVE S KETCHING Sarah Fox

2. T HINGS YOU WILL BE FINDING WHEN DOING CURVE SKETCHING X-intercepts Vertical asymptotes Horizontal asymptotes First derivatives Sign lines Graphs of functions Graphing functions using properties of their derivatives

3. H OW TO FIND X - INTERCEPTS The purpose of finding x-intercepts is so that you will know where your graph crosses the x-axis. To do this you need to solve for x in the numerator of the function you are going to graph. Say your numerator is x 2 – 4 You will then solve for x by setting x 2 – 4 equal to zero So, your x-intercepts with be x= 2 and x= -2

4. H OW TO FIND VERTICAL ASYMPTOTES Vertical asymptotes are important to find because they are vertical boundaries that your graph cannot pass. To find vertical asymptotes you have to solve for x in the denominator in the same way that you found x-intercepts: Set the denominator equal to zero and solve for x.

5. H OW TO FIND HORIZONTAL ASYMPTOTES Horizontal asymptotes are important to find because they are horizontal boundaries that your graph can only pass through once. The three rules on finding horizontal asymptotes are: If the power on bottom is greater then y= 0 If the power on top is greater the asymptote is oblique If the powers are equal use a ratio of the coefficients

6. E XAMPLES OF HORIZONTAL ASYMPTOTES You’re given the function f(x) = _x 2 _ x 2 + 3 This function would follow the third rule, the ratio of coefficients because the powers in the numerator and denominator are both squares, so they are equal. Since there are “invisible” ones in front of both of the x 2, the ratio would be 1 over 1 so the horizontal asymptote would be y=1

7. E XAMPLES OF HORIZONTAL ASYMPTOTES 2 Say you were given the function f(x) = _2x_ x 2 - 1 This function would follow the first rule because the power on bottom is greater so the horizontal asymptote would be y=0

8. E XAMPLES OF HORIZONTAL ASYMPTOTES 3 You are given the function: f(x) = _x 2 - 1_ x Since the power on top is greater, the horizontal asymptote is oblique, and you will have to do long division to find what y equals. You will divide x 2 – 1 by x, and you will find that x goes into x 2 – 1 “x” times with a remainder of -1 That will make your horizontal asymptote y=x

9. F IRST DERIVATIVES After finding your x-intercepts, vertical asymptotes, and horizontal asymptotes, the next step in curve sketching is taking the first derivative of the function you are given. The purpose of taking the first derivative is so that you can solve for x in both the numerator and denominator so you can make what is called a sign line.

10. F IRST DERIVATIVES CONT. Say you are given the function: f(x) = __x__ x 2 + 1 You will use the quotient rule (bottom x derivative of the top + top x derivative of the bottom all over bottom squared) to find the first derivative After you use the quotient rule you should have: f ’(x) = __(x 2 + 1) (1) – (x) (2x)__ (x 2 + 1) 2

11. F IRST D ERIVATIVE C ONT. After you get: f ’(x) = __(x 2 + 1) (1) – (x) (2x)__ (x 2 + 1) 2 You will have to simplify You should then have: f ’(x) = __(1 - x) (1 + x)__ (x 2 + 1) 2 The things you would then use for your sign line are: (1 – x), (1 + x), (x 2 + 1), and (x 2 + 1)

12. H OW TO MAKE A SIGN LINE Sign lines are important because they tell you the integrals for which your graph is increasing or decreasing from the first derivative Using the last example: f ’(x) = __(1 - x) (1 + x)__ (x 2 + 1) 2 When making your sign line, things in the numerator will use darkened circles, and things from the denominator will used open circles

13. S IGN LINES CONT. f ’(x) = __(1 - x) (1 + x)__ (x 2 + 1) 2 (make sign line)

14. G RAPHING Now you will have to put all of the stuff you found (x-intercepts, vertical asymptotes, horizontal asymptotes, and sign lines) together to form your graph Here’s an example: f(x) = __x__ 2x – 1 First, you will need to find the x-intercepts, which you should find to be X = 0

15. G RAPHING CONT. f(x) = __x__ 2x - 1 Then, you will need to find the vertical asymptotes by solving for x in the denominator You should find it to be x = 1/2 Next you will find the horizontal asymptote by going through the horizontal asymptote rules You should find the horizontal asymptote to be y = 1/2 since it’s a ratio of coifficients

16. G RAPHING CONT f(x) = __x__ 2x - 1 Then you will have to take your first derivative so you can make a sign line. You should get your first derivative to be: f ’(x) = __-1__ (2x – 1) 2

17. G RAPHING CONT. f’(x) = __-1__ (2x – 1) 2 Next, the things that will go on your sign line should be: -1, 2x – 1, and 2x – 1 (make sign line)

18. G RAPHING CONT. f(x) = __x__ x- int: x = 0 2x – 1 v. a: x = 1/2 h. a: y = 1/2

19. T RY M E P ROBLEM 1 f(x) = _2x 2 - 18_ x 2 - 4

20. T RY M E P ROBLEM 1 SOLUTION f ’(x) = __20x__ x- int: x = 3 x= -3 (x – 2) 2 (x + 2) 2 v. a: x = 2 x = -2 h. a: y = 2

21. T RY M E P ROBLEM 2 f(x) = __x__ (x 2 + 2) 1/2

22. T RY M E P ROBLEM 2 S OLUTION f ’(x) = __2__ x- int: x = 0 (x 2 + 2) 3/2 v. a: none h. a: y = 1 y = -1

23. T RY M E P ROBLEM 3 f(x) = __x__ (x 2 - 1)

24. T RY M E P ROBLEM 3 S OLUTION f ’(x) = __- (1 + x 2 )__ x- int: x = 0 (x - 1) 2 (x + 1) 2 v. a: x = 1 x = -1 h. a: y = 0

25. H OW TO G RAPH FUNCTIONS USING PROPERTIES OF THEIR DERIVATIVES Three rules to know when graphing functions from their derivatives are: If the graph of the first derivative is above the x-axis the function is increasing. If it’s below the x-axis, the function is decreasing. X-intercepts of the first derivative are max’s and min’s of the function. Max’s and Min’s of the second derivative are inflection points of the function. This shows where concavity changes.

26. T ERMS TO KNOW Concavity: shows the shape of the graph. Concave up “holds water” Concave down “spills water” A point of inflection, or inflection point, occurs where the concavity changes direction A “max” is the highest point on the graph A “min” is the lowest point on the graph

27. H OW TO … CONT. Example:

28. 1985 AB 6 P ART A Part A asks for where the relative max’s and min’s are, and following rule # 2, you should know that x-intercepts of the derivative are max’s and min’s of the function. And you see on the graph that that the x-intercepts are x = -2 and x = 0

29. 1985 AB 6 P ART B Part B asks where on the graph is the function concave up. We know from rule # 3 that: max’s and min’s of the second derivative are inflection points of the function. This shows where concavity changes. This is a good point to make a chart marking each spot where the graph is concave up and concave down. [ -3, -1) concave down ( -1, 1) concave up ( 1, 2) concave down ( 2, 3] concave up

30. 1985 AB 6 P ART C Part C asks for a graph of the function Now you have to use all of the information you found out from the graph of the derivative like: [-3, -2) increasing [-3, -1) c.d. ( -2, 0) decreasing ( -1, 1) c.u. ( 0, 3] increasing ( 1, 2) c.d. ( 2, 3] c.u.

31. 1985 AB 6 P ART C (graph function)

32. T HE E ND © Sarah Fox 2011