Question

what are the characteristics of exponential functions?

graph each function
f(x) = 3^x

write each exponential function.

x f(x)
0 4
1 2
2 1
3 1/2
4 1/4

x f(x)
0 3
1 6
2 12
3 24
4 48

Answer 1

Characteristics of exponential functions (f(x) = A·b^x or f(x) = a·b^x + c)
- General form: f(x) = A·b^x (or with shift f(x) = A·b^x + C). A is initial value (y-intercept), b is the base.
- Domain: all real numbers (−∞, ∞).
- Range: if A>0 and no vertical shift, (0, ∞). If A<0, (−∞, 0). With vertical shift C, range = (C, ∞) or (−∞, C).
- Y-intercept: f(0) = A (since b^0 = 1).
- Horizontal asymptote: y = 0 (or y = C if vertically shifted).
- Monotonicity: if b > 1 the function is increasing (exponential growth); if 0 < b < 1 it is decreasing (exponential decay).
- Smooth and continuous for all x.
- Always nonzero (no x-intercepts) if there is no vertical shift.
- Concavity: if A·b^x > 0 the graph is concave up (second derivative ∝ ln(b)^2 > 0).

Graphing f(x) = 3^x (description and key points)
- Horizontal asymptote: y = 0.
- Increasing (growth) because base 3 > 1.
- y-intercept: (0, 1).
- Key points:
x = −2 → f(x) = 3^(−2) = 1/9
x = −1 → f(x) = 1/3
x = 0 → f(x) = 1
x = 1 → f(x) = 3
x = 2 → f(x) = 9
- Domain: (−∞, ∞). Range: (0, ∞).
- If you sketch it: passes through (0,1), rises slowly for negative x (approaches 0), then rises rapidly for positive x.

Write each exponential function (find A and base b from the tables)

1)
x: 0,1,2,3,4
f(x): 4,2,1,1/2,1/4
- f(0) = A = 4.
- f(1) = A·b = 2 ⇒ b = 2/4 = 1/2.
- Check: 4·(1/2)^2 = 1, etc.
Answer: f(x) = 4·(1/2)^x

2)
x: 0,1,2,3,4
f(x): 3,6,12,24,48
- f(0) = A = 3.
- f(1) = A·b = 6 ⇒ b = 6/3 = 2.
- Check: 3·2^2 = 12, etc.
Answer: f(x) = 3·2^x

If you want, I can produce plots you can download or show a more detailed sketch for each.