A *Continuous Curve* is one where the curve has no "breaks" in it, in other words you could draw the entire curve without taking your pencil off the paper. An example of a curve which is **not** continuous is $y=\frac 1 x$. The picture below shows a continuous curve with 3
**zeros** (where the curve crosses the *x* axis), and 3 **points with zero gradient** (where the curve is momentarily horizontal).

Sketch possible curves that satisfy the following conditions:

- A continuous curve with exactly one point with zero gradient and exactly two zeros.
- A continuous curve with no points with zero gradient and exactly one zero.
- A continuous curve with exactly one point with zero gradient and no zeros.
- A continuous curve with exactly two points with zero gradient and exactly three zeros.
- A continuous curve with exactly two points with zero gradient and exactly two zeros.
- A continuous curve with exactly one point with zero gradient and exactly one zero.
- A continuous curve with exactly two points with zero gradient and exactly one zero.
- A continuous curve with exactly two points with zero gradient and no zeros.

Is there a number of zeros and a number of points with zero gradient for which a continuous curve cannot be drawn? Or is it possible to draw a continuous curve with any number of zeros and any number of points with zero gradient?

### Extension

Can you give examples of algebraic equations that satisfy each of the conditions above? You might like to use Desmos to try out some possible equations.