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# What's That Graph?

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Age 14 to 18

Challenge Level

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Rishik K started from the eight processes, and matched each one to an equation and type of graph. In some cases, Rishik also suggested what the numbers A, B and C could be. Here is Rishik's work:

- A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature - time graph look like?

$$y = \frac{A}{e^x}+B$$

- The height of the valve on a bicycle tyre above the ground is measured after each centimetre that the bicycle travels forwards. What would the height - distance graph look like?

$$y=A\sin {Bx} +A$$

A=12

*If $A=12$ then the highest point above the ground that the valve reaches is 24 centimetres. That means the bicycle has very small wheels!*

- I throw a tennis ball straight up into the air and catch it. The height of the ball from the ground is measured over the time of the journey using freeze-frame photography. What would the height - time graph look like?

$$y=Ax^2$$

A has to be a negative number.

- I measure several objects using inches and then using metres, plot them on a scatter graph, and join the points. What would the metres - inches graph look like?

$$y=Ax$$

A definitely has to be more than 0.

- I jump out of a plane and the distance fallen from the plane is measured every $0.1$ second until I open my parachute. What would the distance fallen - time graph look like whilst in freefall?

$$y=Ax^2+Bx+C$$

A must be more than 0. If A was negative, it would instead be an upside-down parabola. I think B and C may be 0.

- I drive at 70 miles an hour along the motorway and note the reading on my odometer (mile counter) every 5 minutes. What would the odometer reading - time graph look like?

$$y=Ax+B$$

A is an incredibly small positive fraction and B is 1 because the line has a y intercept at (0,1)

*It looks like this conclusion comes from looking at the graph - but it's important to think about the real-life process as well, especially since we don't know the scale on the graph. Is it possible that the graph here could be swapped with the graph for process 4 (measuring objects in inches and in metres)?*

- I blow up a roughly spherical balloon using a balloon pump. After each pump I measure the radius of the balloon in centimetres. What would the radius - number of pumps graph look like?

$$y=\sqrt[3]{x}$$

- I suck water through a straw out of a large beaker at a constant rate and measure the volume of liquid remaining at various times. What would the volume-time graph look like?

$$y=Ax+B$$

A is negative and B is 5 because the line crosses the y axis at (0,5).

*Again, remember that the graph scale is not given, so the value of the intercept cannot be read from the graph.We also do not know which line represents the x- or y- axis.*