Topic 5: Graph Sketching and Reasoning

Topic 5: Graph Sketching and Reasoning Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 5th August 2013

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. ?  For multiple choice questions (e.g. SMC), click your choice to reveal the answer (try below!) Question: The capital of Spain is:  A: London  B: Paris  C: Madrid

ζ Topic 5 – Graph Sketching and Reasoning Part 1: Sketching Fundamentals

Graph Features? What happens either side of undefined values of ? ? y-intercept? ? y as ? ? y as ? ? Turning Points? ? Roots? ? Asymptotes? ? An asymptote is a straight line that a curve approaches at infinity.

Graph Features? You may not have covered the following terminology yet: -1 The domain of the function is the possible values of the input: ? The range of the function is the possible values of the output: ? The roots of the function are the inputs such that the output is 0: The roots are also known as ‘zeros’ of the function, but never as the ‘x-intercepts’!

The two main ways of sketching graphs 1 Thing about the various features previous discussed. And/or consider the individual components of the function separately, and think how they combine. 2 We’re multiplying these two individual functions together. What happens as increases? What happens at the peaks and troughs of the sin graph? Try sketching it!

? Start say with . Whenever , then . And whenever , then . Notice also that when is negative, multiplying by causes the graph to be flipped on the -axis.

? Think about what happens when we add the graphs and Notice the asymptote . Non-vertical/horizontal asymptotes are known as oblique asymptotes.

? This is similar to the last, except we’re using and to work out the peaks and the troughs. The interesting question is what happens when . Let’s explore this...

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: When we were evaluating for , we get , 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 and when tends towards some value , then: So for our example, we can’t evaluate directly when , but using the rule: Nice! Note that differentiates to .

Values to a power On the same axis, sketch and . ? It’s easy to forget that for values in the range , the higher the power we raise it to, the smaller it becomes. Exercise caution!

Values to a power Draw a graph of . On the same axis, draw Notice that: (a) a squared value is always positive and (b) when , we’ll find that .

3D graphs Sketch (where and are on a horizontal plane and points upwards) ? Explanation: We know that is the equation of a circle (where is a constant). Thus for a fixed value of we have a circle (with radius , thus must be positive). As increases, the radius increases, and we get gradually get larger circles. ?

Composite functions Sketch ? As increases, increases more rapidly. We’re effectively ‘speeding up’ across the sin graph, so the oscillation period gradually decreases. Note also that since , we have symmetry across the -axis.

Composite functions Sketch ? As increases, increases less rapidly. The input to the sin function increases less rapidly, so we move across the sin graph more slowly. Note also that the domain is .

Reciprocal functions Sketch Here’s a sketch of . What will its reciprocal look like? and , so the graphs touch for leads to asymptotes.

Functions transforms Sketch Hint: If we start with , how could we transform this to get the desired function? Click to view transformation ? 1 Click to view transformation -2 ?

Putting it all together… Because of multiplication by , peaks gradually become shallower. Because of (and for all ), always positive. The inside the causes the oscillation period to decrease. We could have eliminated this choice by trying .  A  B  C  D [Source: Oxford MAT 2007]

y2 = (x-1)/(x+1) • For this graph, it might be helpful to think about: • How do deal with the y2. • The asymptotes (both horizontal and vertical). • The domain of x (determine this once you’ve dealt with the y2). • 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.

ζ Topic 5 – Graph Sketching and Reasoning Part 2: Reasoning about Solutions

Polynomials A polynomial expression is of the form: where are constants (which may be 0). Note that the powers of the variable must be positive (or 0). is not a polynomial expression as there’s a power of . The degree of a polynomial is the highest power. Names of polynomials: A polynomial of degree 4. ? ? ?

Polynomials The range of a quadratic however is finite, as it has a maximum/minimum (depending on whether the coefficient of is negative or positive) The range of a cubic is to , i.e. the entirety of . Can we generalise this to polynomials of any degree?

Polynomials Polynomials of even degree will always have a finite range with a minimum or maximum. It will be ‘valley’ shaped if the coefficient of the highest-power term is positive. Polynomials of odd degree will always have a range which spans the whole of the real numbers. It goes ‘uphill’ if the coefficient of the highest-power term is positive.

Number of Roots What can we therefore say about the potential number of roots? (where the polynomial has degree ) Odd Degree Even Degree ? Minimum Roots: ? ? ? Maximum Roots:

Number of Roots We can shift a graph up and down by changing the constant term. i.e. the in Click to Start Animation Number of Distinct Roots: 1 2 3 4 0 1 2 3 4 0

Number of Roots Question: Sketch 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: STEP 1 (2012)] By factorising, . This is a quartic, where is always positive, and has repeated roots at : By changing , we shift the graph up and down. Then we can see that: 0 roots: When 1 root: Not possible. 2 roots: When 3 roots: 4 roots: a) ? b) ? ? ? ? ?

Repeated factors/roots Sketch: ? • In general: • When appears once as a factor:Line crosses-axis at • When is a repeated factor:Line touches-axis at • When is a doubly-repeated:Point of inflection on -axis at ? ?

Repeated factors/roots Here’s a challenge! Sketch: ? Inflection point Touches axis Crosses axis The polynomial is of degree 6. So there’s up to 5 turning points. 2 are ‘used up’ by the inflection point. There’s 1 turning point at the origin. So that leaves 2 turning points left. We can therefore deduce one turning point appears somewhere in and the other in .

Turning Points Polynomials’ turning points oscillate between or max max min min ? For a polynomial of order , the maximum number of turning points is:

Points of Inflection However, when we have a point of inflection, then two of the turning points effectively ‘conflate’ into one. It’s a bit like having a max point immediately followed by a min (or vice versa) (In fact we can have more than 2 turning points ‘conflate into one’. Consider . A quartic can have up to 3 stationary point, but this graph only has 1!)

Points of Inflection A point of inflection is where the curve changes from curving downwards to curving upwards (or vice versa). It may or may not be a stationary point: (depending on whether ) Non-stationary point of inflection Known as a ‘non-stationary point of inflection’ Stationary point of inflection Known as a ‘saddle-point’

Strategies for determining number of solutions METHOD 1: Reason about the graph • As we already have done: • If it’s a polynomial, is the degree even or odd? • If we have a constant that can be changed, consider the graph shifting up and down. • We may have to find the turning points (by differentiation or completing the square) METHOD 2: Factorise (when possible!) e.g. This cubic conveniently factorises to: We can see it has three solutions (two of them equal). Look out for the difference of two squares!!! ? METHOD 3: Consider the discriminant Remember that if we have a quadratic , there are real solutions if: ?

Example [Source: Oxford MAT 2009] B  D  A   C

Example • By differentiating to find the turning points: So the turning points occur at . Then considering the graph of the quartic: If the x-axis is anywhere in the horizontal trip between the maximum and the greater of the two minimums (whichever it is), we’ll have four solutions because the line will cross the axis 4 times. The y-values of the turning points are and respectively. So so the maximum is above the x-axis, and so that the greater of the two minimums occurs below the x-axis.

Example [Source: Oxford MAT 2009]  A  B  C  D

Example • Spot when you can use the difference of two squares • Make use of the discriminant. • Thus • If , and using the discriminant on the first, . • Using the discriminant on the second

Topic 5: Graph Sketching and Reasoning