exponential 2 ref points. (0,1) and (1,b) when b^x formula ab^(x-h)+k ref point (0,1) is (h, a+k) Ref point 2 is (h+1,ab+k). How is this formula derived ?

Question

exponential 2 ref points. (0,1) and (1,b) when b^x formula ab^(x-h)+k ref point (0,1) is (h, a+k) Ref point 2 is (h+1,ab+k). How is this formula derived ?

Answer 1

This question can best be understood through the use of function transformations. Simply put, each type of function is called a family, and each family has a parent. The function family you are working with is the exponential functions, and the parent is y = b^x. Every other member of the family can be expressed using some combination of stretches, flips, and shifts of the parent.

Let's use the example you presented: y = a•b^{x - h} + k. You have three transformations present: a vertical stretch (a), a horizontal shift (h), and a vertical shift (k). Vertical stretches are multipliers that only multiply the y-values of the parent function. Horizontal shifts will only affect the x-values of the function and will add or subtract. Vertical shifts will only affect the y-values of the function and will add or subtract. Following order of operations, the multiplier will come first (a), then the shifts (h and k).

So, if we apply those transformations in order to the parent function points, we see the following:

(0, 1) becomes (0, a) and (1, b) becomes (1, ab) when we apply the vertical stretch. Then,

(0, a) becomes (0 + h, a) and (1, ab) becomes (1 + h, ab) when we apply the horizontal shift. Finally,

(0 + h, a) becomes (0 + h, a + k) and (1 + h, ab) becomes (1 + h, ab + k) when we apply the vertical shift.

Some simple rearrangement of the final points will get you the reference points in the format you've given.