Question

Which exponential function has a growth factor of 5?

f(x) = 2(5x)
On a coordinate plane, an exponential function decreases from quadrant 2 into quadrant 1 and approaches y = 0. It goes through (negative 1, 2) and crosses the y-axis at (0, 0.5).
f(x) = 0.5(2x)
A 2-column table has 5 rows. The first column is labeled x with entries negative 2, negative 1, 0, 1, 2. The second column is labeled f (x) with entries one-eighth, one-fourth, one-half, 1, 2.

Answer 1

I'll correct and write each exponential function properly, with brief work.

1) "Which exponential function has a growth factor of 5?
You wrote f(x) = 2(5x)."
- Growth factor = base b in a·b^x. The correct exponential with growth factor 5 and initial value 2 is
f(x) = 2·5^x.
- Note: 2(5x) means 10x (a linear function), not exponential.

2) "Decreasing from Q2 to Q1, approaches y = 0, passes through (-1, 2) and (0, 0.5). You wrote f(x) = 0.5(2x)."
- Use f(x) = a·b^x. From (0,0.5) we get a = 0.5.
- From (-1,2): 0.5·b^{-1} = 2 → (0.5)/b = 2 → 1/b = 4 → b = 1/4.
- So the exponential is f(x) = 0.5·(1/4)^x (a decay).
- Again, 0.5(2x) is linear if interpreted as multiplication; 0.5·2^x would be increasing, so it's not the right model.

3) Table x = -2,-1,0,1,2 with f(x) = 1/8,1/4,1/2,1,2
- Values double each step, so base b = 2 and a = f(0) = 1/2.
- The function is f(x) = (1/2)·2^x (equivalently 2^{x-1}).