Exponential graphs and parameters

For y = abˣ with a > 0 and b > 0, b ≠ 1, the vertical intercept is a. The factor b multiplies the output when x increases by one. The graph increases when b > 1 and decreases when 0 < b < 1. In y = ab⁻ˣ the step multiplier is 1/b. Positive outputs approach zero in one direction without reaching the horizontal axis.

To recover parameters, start with a point at x = 0 if one is given. Divide two nonzero outputs to remove a and relate the ratio to the input difference. Keep exact fractions throughout. A sketch should show the intercept, increasing or decreasing behaviour and the approach to the axis; it must not turn into a straight line merely because only two points are supplied.

Worked example

The graph y = ab⁻ˣ, with a > 0 and b > 1, passes through (0,3) and (2,3/4). Find a and b and describe the graph.

1. At x = 0, y = a, so a = 3.
2. At x = 2, 3/b² = 3/4, hence b² = 4. The condition b > 1 selects b = 2.
3. The output halves when x increases by one, and remains positive for every real input.

Answer: a = 3, b = 2; the decreasing graph crosses the vertical axis at 3 and approaches y = 0 as x increases.

Revise first: Fractional and negative indices.