Combining Functions

Package of problems

Things you may have noticed

Sketch the graph of $y=\dfrac{1}{x+a}$ for different values of $a$. We suggest you start by trying the values $a=0$, $1$, $-1$.

What do you notice? $y=\dfrac{1}{x+1}$

The graph of $y=\dfrac{1}{x}$ is familiar. The family of graphs of $y=\dfrac{1}{x+a}$ all look very similar.

Can you describe how they are related and the significance of the value $a$?

Now sketch the graph of $y=\dfrac{1}{x^2+a}$ for different values of $a$, again starting with $a=0$, $1$, $-1$.

What do you notice this time?

Can you explain why the graphs behave in this way?

The shape of these graphs is very different for different values of $a$. We can split our graphs into the cases $a<0$, $a=0$ and $a>0$.

When $a=0$ $y=\dfrac{1}{x^2}$

This is the familiar graph $y=\dfrac{1}{x^2}$.

We can visualise it as the reciprocal of $y=x^2$.

• When $x=0$, $x^2=0$ so $y=\frac{1}{x^2}$ has a vertical asymptote.
• As $x$ increases, $x^2$ increases so $\frac{1}{x^2}$ decreases towards zero.
• The function is even since the $x$ value is squared, so the $y$-axis is a line of symmetry.

When $a<0$ $y=\dfrac{1}{x^2-1}$

We might have expected $y=\dfrac{1}{x^2-1}$ to be a translation of $y=\dfrac{1}{x^2}$, but instead it does something much more interesting!

To understand this behaviour, we could think about the graph of $y=x^2-1$ and what the reciprocal will look like, paying particular attention to where it crosses the $x$-axis. How, then, does the shape of $y=\dfrac{1}{x^2+a}$ change as $a$ takes negative values other than $-1$?

Another way to visualise this is to rewrite the function in terms of other functions whose graphs we are familiar with. Since $a$ is negative, we could rewrite our function as $y=\dfrac{1}{x^2-b^2}$ (where $b^2=-a$). Can you now rewrite this further?

When $a>0$ $y=\dfrac{1}{x^2+1}$

Now, we find that there are no vertical asymptotes. Can you explain why by thinking about the graph of the quadratic $y=x^2+1$?

What if we rewrite this function as $y=\dfrac{1}{x^2+b^2}$? Can you expand this as we did above?

In this case, when $a>0$, what happens to the graph as $a$ takes different positive values?

Joining them up

As $a$ gets close to zero, can you visualise how each of these more interesting graphs will become closer and closer to the graph of $y=\dfrac{1}{x^2}$?

Use graphing software or the applets in the next section to view this transition.