Topic 4: Graph Sketching

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Topic 5: Graph SketchingDr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 18th July 2013

Slides not yet completeThese 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.

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? 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: London B: Paris C: Madrid

For multiple choice questions (e.g. SMC), click your choice to reveal the answer (try below!)

Question: The capital of Spain is:

Graph features?Asymptotes?

𝑦=ln (𝑥 )𝑥

Turning Points?

As ?

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. 𝑥>0 𝑦 ≤ 1𝑒

To differentiate use the quotient rule.This gives . So

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? ?

?As ?

As x becomes small, y tends twoards

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TO EDITStudents are very unlikely to have

covered ln!

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.

𝑦=ln(𝑥 )

1

2

𝑦=𝑥

Then how do we sketch ?Think about what happens when we divide by y values of each function.Here we’re

going to have 0 when we divide 0 by 1.

After x=1, x is increasing faster than , so the rate at which increases gradually decreases. Eventually this rate of increase shrinks to 0.

As x > ln(x) for any real x, tends towards 0 as .

As x becomes small, we’re dividing by an increasingly small number <1. So decreases more rapidly than .

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.

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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...

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Indeterminate FormsWe’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:

00 00

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.

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!

Other sketches

Now consider what happens with the following:

ex

x__x__sin(x)

sin(√x)

sin( )1x esin x

To confirm your sketches, you can using www.graphsketch.com or type “sketch [my equation]” into www.wolframalpha.com

x + 1x

Hint: This has two asymptotes, one of them diagonal.

x / sin x

sin(√x)

y = sin(x)y= sin(√x)

ex / x

sin(1/x)

𝒚=𝒙

𝒚=𝒙+𝟏𝒙

Notice the asymptote .

esin x

𝟏𝒆

e

y2 = (x-1)/(x+1)

For this graph, it might be helpful to think about:

1. How do deal with the y2.2. The asymptotes (both horizontal and vertical).3. The domain of x (determine this once you’ve

dealt with the y2).4. Roots.

y2 = (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.

Reasoning about numbers of solutionsShifting a graph vertically up and down often results in a changing number of solutions/roots. Just use the shape of the graph to reason.

Question: For what values of does the equation have the following number of distinct roots (i) 0, (ii) 1, (iii) 2, (iv) 3, (v) 4. [Source: Follow up Q on STEP 1 (2012)](Hint: first sketch the graph for a particular value of for which the equation factorises)

When , then . This is a quadric, where y is always positive, and has repeated roots at :

By changing , we shift the graph up and down. Then we can see that:

i) 0 roots: When ii) 1 root: Not possible.iii) 2 roots: When iv) 3 roots: v) 4 roots:

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To edit: I’ve got some better stuff I’ve made for this subtopic in the MAT slides