A. Linear functions grow at a constant rate. This means that as you increase the input (x-value), the output (y-value) increases by the same amount every time. For example, if a linear function grows by 2 for every increase of 1 in x, then moving from x=1 to x=2 will increase the output by 2, and moving from x=2 to x=3 will again increase the output by 2.
Exponential functions, on the other hand, grow at an increasing rate. This means that as you increase the input, the output increases by a factor. For instance, if an exponential function doubles its output as x increases, then from x=1 to x=2, the output might go from 2 to 4, and then from x=2 to x=3, it goes from 4 to 8. The rate of increase gets larger as x increases.
B. The statement “No matter what the growth rate of a linear function is, that of an exponential function will always exceed it” means that, over time, exponential functions will outpace linear functions regardless of how fast the linear function is growing.
For example, if your linear function increases by 2 each time and your exponential function doubles its previous output, over time, the exponential function will start to grow much faster.
If we look at specific values:
- At x=1, the linear function might be at 2, while the exponential function (starting at 1) would be at 2.
- At x=2, the linear function would be at 4, but the exponential function jumps to 4.
- By x=3, the linear output is at 6, but the exponential function shoots up to 8.
As you go further, the gap between the two outputs widens significantly, demonstrating that, while linear growth adds the same amount each time, exponential growth multiplies, leading to much larger results over time.