Robert was having trouble visualizing the graphs of exponential functions and the ways an exponential function can be generalized (the characteristics that every function of the form f(x) = a * b^x will share). I started by drawing a few diagrams for him:
Robert was able to understand that in a function f(x) = b^x, regardless of the base b, when x is 0, the function has a value of 1. This is because any base raised to an exponent of 0 is equal to 1.
I asked Robert to consider how the y-intercept would change if the function b^x times a constant a. First I gave him an example of f(x) = 2 * b^x. He told me that the y-intercept is (0, 2). I then asked him to extend this idea to any constant a. He was able to see that the y-intercept would be (0, a).
I then asked Robert to consider the range of a function f(x) = b^x. He was able to see that the function will never reach 0, or be less than 0. I then asked him to consider what would happen if he added five on the end, as in the function g(x) = b^x + 5. After the previous question, Robert quickly concluded that the range of the function is all numbers greater than 5.