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# Curve Hunter

##### Age 14 to 18Challenge Level

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:

1. A continuous curve with exactly one point with zero gradient and exactly two zeros.
2. A continuous curve with no points with zero gradient and exactly one zero.
3. A continuous curve with exactly one point with zero gradient and no zeros.
4. A continuous curve with exactly two points with zero gradient and exactly three zeros.
5. A continuous curve with exactly two points with zero gradient and exactly two zeros.
6. A continuous curve with exactly one point with zero gradient and exactly one zero.
7. A continuous curve with exactly two points with zero gradient and exactly one zero.
8. 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.