### Combining Functions

Package of problems

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## Suggestion

### Graph sketching

When sketching the graph of an unfamiliar function, it might help to think about the following.

• Where does the graph cross the axes?
• When is the function positive and when is it negative?
• Does it have any asymptotes?
• Is it increasing or decreasing?
• What happens as $x$ gets very large?
• Is it related to any other graphs whose shape I know?

As a check, you could substitute in some values of $x$ such as $x=0$, $1$, $2$, $-1$.

If you want a final check of your sketches, you could use a graphical calculator or software such as Desmos. Alternatively, you could use the Interactive graphs section of this resource.

Now sketch the graph of $y=\dfrac{1}{x^2+a}$ for different values of $a$.

In addition to the above, you might find it helpful to first sketch the graph of $y=x^2+a$ (for your chosen value of $a$). How does the graph of $y=\dfrac{1}{f(x)}$ relate to the graph of $y=f(x)$?