1. Topic 4: Graph Sketching Dr J Frost (jfrost@tiffin.kingston.sch.uk)

2. Topic 5: Graph Sketching Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 18th July 2013 Slides not yet complete These slides will be ready for September 2013. I’ve uploaded the partially complete ones for the purposes of L6 people trying to prepare for Oxbridge entry over the summer.

3.   Slide guidance ? Any box with a ? can be clicked to reveal the answer (this works particularly well with interactive whiteboards!). Make sure you’re viewing the slides in slideshow mode. A: LondonB: ParisC: Madrid For multiple choice questions (e.g. SMC), click your choice to reveal the answer (try below!) Question: The capital of Spain is:

4. Graph features? Asymptotes? Turning Points? Roots and y-intercepts?Domain and Range? x = 0 and y = 0 As x becomes large, y tends towards 0. Has a root x = 1. ? ? ? ? ? ? TO EDIT Students are very unlikely to have covered ln!

5. The two main ways of sketching graphs: Use the various features previous discussed. And/or consider the individual components of the function separately, and think how they combine. 1 2 Here we’re going to have 0 when we divide 0 by 1.

6. y = x sin(x) Start say with y = sin(x). Whenever sin(x) = 1, then x sin(x) = x. And whenever sin(x) = -1, then x sin(x) = -x. Notice also that when x is negative, multiplying sin(x) by x causes the graph to be flipped on the y-axis. ?

7. y = sin(x) / x This is similar to the last, except we’re using y = 1/x and y = -1/x to work out the peaks and the troughs. The interesting question is what happens when x = 0. Let’s explore this... ?

8. Indeterminate Forms We’re used to seeing divisions by 0 leading to vertical asymptotes. But there’s nothing mathematically problematic about this: we just shoot to +∞ Or -∞. However, there are some divisions and other expressions which are quite simply, have no value. These are known as indeterminate forms: 0000 0 When we were evaluating for x = 0, we get 0/0, which is indeterminate. To evaluate it, we need to use something called l’Hôpital’s rule.

9. l’Hôpital’s Rule If you want to evaluate, but both f(x) = 0 and g(x) = 0 when x tends towards some value c, then: So for our example, we can’t evaluate sin(x) / x directly when x = 0, but using the rule: Nice!

10. Other sketches Now consider what happens with the following: exxexx __x__ sin(x) sin( √ x) sin( ) 1x1x e sin x To confirm your sketches, you can using www.graphsketch.com or type “sketch [my equation]” into www.wolframalpha.comwww.graphsketch.comwww.wolframalpha.com x + 1x1x Hint: This has two asymptotes, one of them diagonal.

11. x / sin x

12. sin( √ x) y = sin(x) y= sin( √ x)

13. e x / x

14. sin(1/x)

16. e sin x e

17. y 2 = (x-1)/(x+1) For this graph, it might be helpful to think about: 1.How do deal with the y 2. 2.The asymptotes (both horizontal and vertical). 3.The domain of x (determine this once you’ve dealt with the y 2 ). 4.Roots.

18. y 2 = (x-1)/(x+1) Not defined for -1<x<1. Repeated above and below x-axis because we have y =  √... As x becomes larger, the +1 and -1 has increasingly little effect, so y =  1 for large x.

19. Reasoning about numbers of solutions Shifting a graph vertically up and down often results in a changing number of solutions/roots. Just use the shape of the graph to reason. ? To edit: I’ve got some better stuff I’ve made for this subtopic in the MAT slides