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# Coordinates and Graphs : Further Mathematics (AS-Level) : Mathematics Content Knowledge

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Further Mathematics (AS-Level)
Coordinates and Graphs
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# 1. How confident are you that

## Example

Before it is possible to sketch a graph there are five aspects of the behaviour of the function that need to be ascertained.

Once these five aspects have been found then a sketch of the graph should be a straightforward process.

1. Symmetry

Consider whether the function is odd or even – in order to find the appropriate level of symmetry. An even function has reflective symmetry in the y-axis. Even functions are such that $f(-x)=f(x)$. This can be discovered either by careful consideration of the function itself, or by substitution of a couple of values for$x$.

Odd functions are such that $f(-x)=-f(x)$ and have rotational symmetry of order 2 about the origin (they are unchanged after a half turn about the origin).

2. Asymptotes parallel to the axes. (Note that oblique asymptotes – that is ones that are not horizontal or vertical are not necessary at this level). For vertical asymptotes one must consider whether there are any values for $x$which leave the function undefined. These occur when the denominator of the fraction is equal to zero. Horizontal asymptotes are found by considering whether any values of y are unobtainable despite whatever values for $x$are tried. These may also be found in the fifth section below.

3. Once the asymptotes have been found it is necessary to consider how the graph behaves close to the asymptote.

Consider $f(x)=\frac{1} {x-4}$

There is an asymptote at $x=4$ since that would make the denominator equal to zero. It is now necessary to find the value of $f(x)$ when $x$ is close to 4. Finding $f(3)=-1$ and $f(5)=1$ shows that the graph heads down towards negative infinity on the left hand side of the asymptote, and down from positive infinity on the right hand side of the asymptote.

This sketch illustrates this point:

4. Does the graph meet either axis?  Try solving $f(x)=0$to find where the graph meets the x-axis, and finding $f(0)$ will show where the graph meets the y-axis.

5. What happens for small values of $x$ and large values of $x$? Small values of $x$ are only necessary when the function is undefined when $x=0$.
For example: $f(x)=\frac{(x+1)(x+2)} {x}$

If we expand the brackets and separate into three terms then we have $f(x)=x+3+\frac{3} {x}$

So away from $x=0$ when the $\frac{3} {x}$ terms dominates $f(x)$ then the $x+3$ part is the most significant. So as the function approaches the origin it roughly looks like $x+3$.

Similarly it is also necessary to look at the behaviour of the function for large values of $x$, essentially the largest power of $x$ is going to dominate.

For example: $f(x)=\frac{2x+3} {x}$.

This can be written as the sum of two separate fractions, so $f(x)=2+\frac{3} {x}$.

So as $x$ becomes very large, the $\frac{3} {x}$ term becomes very small, and $f(x)$ will head towards the value 2.

This gives us a horizontal asymptote of y=2, and the sign of $\frac{3} {x}$ will tell you whether the graph approaches the asymptote from above or below.

## What this might look like in the classroom

It is necessary to sketch many graphs, and use a variety of examples to illustrate the five points made above. Here are some examples to use for each point:

1. Symmetry

$f(x)=x^2sinx$will be found to be an odd graph.

$f(x)=x^2cosx$will be found to be an even graph.

$f(x)=(x^2-x)sinx$ is neither odd nor even.

(Well worth showing students some which are neither odd nor even, many students assume that all functions have to be either odd or even).

2. Asymptotes

$f(x)=\frac{1} {(x+2)(x-3)}$ has two vertical asymptotes.

$f(x)=4+\frac{1} {x}$ has a horizontal asymptote. (Try solving $f(x)=4$)

3. Graph close to the asymptote

Try this again $f(x)=\frac{1} {(x+2)(x-3)}$ and consider behaviour either side of each asymptote.

4. Graph meeting the axes.

Consider $f(x)=\frac{x+5} {(x+2)(x-3)}$.

5. Small and large values of $x$

$f(x)=4+\frac{1} {x}$ has been met already in (2) but can be considered under this heading as well.

## Taking this mathematics further

Students would do well to consider more complex equations;
such as $f(x)=cos(x^2)$ and $f(x)=\frac{1} {x^3+2x^2-x-2}$ in order to test their skills more fully.

The use of graph plotting software makes this operation easier to introduce but students need to develop strategies for working with complex equations.

## Making connections

Students need to remember that the x-axis is also the line $y=0$, and the y-axis is the line $x=0$. This will help them identify which axis to mark their solutions on. They will need to be able to factorise quadratics, and be able to consider just part of a fraction – for example when looking for the values of $x$ that make the denominator equal to zero.

Students may also need reminding of the concept of sketching – as being different from plotting points.

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