Explain in a variety of ways how you can distinguish the exponential function f(x)=2^x from the quadratic function f(x)=x^2 and linear function f(x)=2x. (Hint: Compare the rate of change using finite differences in table of values; identify a constant ratio in the table of values). (10 - marks)

I would make a sketch of each function on the same grid.

Also they told you what to do in the hint, why not do that?

I do not know how to do the hint.

To distinguish between the exponential function f(x) = 2^x, the quadratic function f(x) = x^2, and the linear function f(x) = 2x, we can use a variety of methods:

1. Analyzing their formulas: The exponential function f(x) = 2^x has a base of 2 raised to the power of x. The quadratic function f(x) = x^2 contains a squared term, and the linear function f(x) = 2x has a linear relationship with x.

2. Graphing the functions: Plotting each function on a graph can help visualize the differences. The exponential function f(x) = 2^x will display a growth that becomes steeper as x increases. The quadratic function f(x) = x^2 will show a U-shaped curve, while the linear function f(x) = 2x will display a straight line with a constant slope.

3. Comparing the rates of change using finite differences: Create a table of values for each function, calculating the change in y values for consecutive x values. In the exponential function f(x) = 2^x, you will observe that the differences between y values increase exponentially. In the quadratic function f(x) = x^2, the differences will exhibit a linear pattern. Finally, the linear function f(x) = 2x will result in constant differences between consecutive y values.

4. Identifying constant ratios in the table of values: Another way to distinguish the functions is to calculate the ratios between consecutive y values for various x values. For the exponential function f(x) = 2^x, the ratios will be constant, indicating exponential growth. In the quadratic function f(x) = x^2, the ratios of the y values will vary, indicating a parabolic curve. In the linear function f(x) = 2x, the ratios will also be constant, but with a different value than the exponential function.

By employing these different methods, we can clearly distinguish between the exponential, quadratic, and linear functions.