I don’t see the graph itself — please either upload the image or tell me two clear points from the graph (for example the y-intercept and one other point). In the meantime, here’s a clear step-by-step method you can use to get f(x)=a(b)^x from a graph, plus a worked example.
- Identify two points on the graph: Preferably use the y-intercept (x=0) and one other point with integer coordinates. Label them (x1,y1) and (x2,y2). If you have the y-intercept, that gives a immediately because a = f(0) = y at x=0.
- If you have the y-intercept (0,a): then a is known. To get b, use another point (x2,y2). From f(x)=a·b^x we have y2 = a·b^{x2}, so
b^{x2} = y2 / a
and therefore
b = (y2 / a)^{1/x2}. - If you don’t have the y-intercept but have two points (x1,y1) and (x2,y2) (with x2 ≠ x1), solve the two equations
y1 = a·b^{x1}
y2 = a·b^{x2}.
Divide the second by the first to eliminate a:
y2 / y1 = b^{x2-x1}
so
b = (y2 / y1)^{1/(x2-x1)}.
Then plug b back into y1 = a·b^{x1} to solve for a: a = y1 / b^{x1}. - Check your result: substitute a and b into f(x)=a·b^x and verify it matches the points used (and ideally another point on the graph).
Worked example (so you can follow the algebra): Suppose the graph passes through (0,3) and (2,12).
- a = f(0) = 3.
- Use (2,12): 12 = 3·b^2 → b^2 = 12/3 = 4 → b = 2 (take the positive root for exponential growth; if graph shows decay you will get 0<b<1).
- So f(x) = 3·2^x. Check: f(2) = 3·2^2 = 12, matches the point.
If you give me the graph or tell me two accurate points (for example the y-intercept and one other point, or two grid points on the curve), I will compute the exact f(x)=a(b)^x for that graph and show the work.