This worksheet guides students through identifying key characteristics of exponential functions, including asymptotes, domain, range, and y-intercepts, using both graphs and tables.

Let’s solve each problem step by step.

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Problem 1: f(x) = 2(3)^x

We’re given the graph and function. Let’s find characteristics:

- Growth or decay?
The base is 3, which is > 1 → exponential growth.

- Asymptote at?
Exponential functions of this form have a horizontal asymptote at y = 0 (the x-axis).

- Domain:
All real numbers — you can plug in any x-value.

- Range:
Since it grows from near zero but never touches it, range is y > 0.

- y-intercept:
Plug in x = 0: f(0) = 2*(3)^0 = 2*1 = 2 → (0, 2)

✔ Verified with graph: curve starts near x-axis on left, goes up through (0,2), steeply rising → matches growth.

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Problem 2: f(x) = 3*(1/2)^x + 4

- Growth or decay?
Base is 1/2, which is < 1 → exponential decay.

- Asymptote at?
The “+4” shifts the whole graph up. Original asymptote for (1/2)^x is y=0, so now it’s y = 4.

- Domain:
Again, all real numbers.

- Range:
Since it decays toward 4 but never reaches it, and starts higher, range is y > 4.

- y-intercept:
Plug in x = 0: f(0) = 3*(1/2)^0 + 4 = 3*1 + 4 = 7 → (0, 7)

✔ Graph shows curve starting high on left, decreasing, leveling off above x-axis around y=4 → matches.

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Problem 3: f(x) = 4^(x−1) − 3

First, fill the table:

| x | f(x) calculation | f(x) value |
|----|------------------------------|----------|
| -2 | 4^(-2−1) − 3 = 4^(-3) − 3 | 1/64 − 3 ≈ -2.984 → but let's use exact fractions or decimals? Actually, better to compute exactly: 4^-3 = 1/64 → 1/64 - 3 = -191/64 ≈ -2.984 → but maybe they want decimal? Wait — let’s do exact values as fractions or simplified decimals.

Actually, let’s compute numerically for clarity:

x = -2: 4^(-3) = 1/(4^3) = 1/64 ≈ 0.015625 → minus 3 → ≈ -2.984 → but perhaps we keep as fraction? Or maybe round? But since other problems use integers, let’s check if we made mistake.

Wait — actually, let’s recalculate carefully:

f(x) = 4^(x−1) − 3

x = -2: exponent = -3 → 4^(-3) = 1/64 → 1/64 - 3 = (1 - 192)/64 = -191/64 → messy. Maybe they expect decimal approximations? But looking at context, probably acceptable to write as decimals rounded to two places? Or maybe I misread.

Wait — let’s try integer outputs where possible.

x = 1: 4^(0) - 3 = 1 - 3 = -2 → nice.

x = 2: 4^(1) - 3 = 4 - 3 = 1

x = 3: 4^2 - 3 = 16 - 3 = 13

x = 0: 4^(-1) - 3 = 1/4 - 3 = -2.75

x = -1: 4^(-2) - 3 = 1/16 - 3 = -2.9375

x = -2: 4^(-3) - 3 = 1/64 - 3 ≈ -2.984 → still messy.

But perhaps the worksheet expects us to leave as fractions or decimals? Since no instruction, I’ll use decimals rounded to two places for simplicity, but note that exact fractions are more precise.

Alternatively, maybe the student is expected to just compute and plot — so let’s list:

Table:

x | f(x)
----|------
-2 | 4^{-3} - 3 = 1/64 - 3 = -191/64 ≈ -2.98
-1 | 4^{-2} - 3 = 1/16 - 3 = -47/16 = -2.9375 ≈ -2.94
0 | 4^{-1} - 3 = 1/4 - 3 = -11/4 = -2.75
1 | 4^{0} - 3 = 1 - 3 = -2
2 | 4^{1} - 3 = 4 - 3 = 1
3 | 4^{2} - 3 = 16 - 3 = 13

Now, characteristics:

- Growth or decay?
Base 4 > 1 → growth.

- Asymptote at?
Horizontal shift doesn’t affect asymptote location vertically; original 4^x has asymptote y=0, then subtract 3 → asymptote at y = -3.

- Domain: All real numbers.

- Range: Since it grows from near -3 upward, range is y > -3.

- y-intercept: When x=0, f(0)= -2.75 → (0, -2.75) or (0, -11/4)

✔ Plotting these points would show curve coming from left near y=-3, passing through (0,-2.75), (1,-2), (2,1), etc. → increasing → growth.

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Problem 4: f(x) = 2*(1/2)^x + 3

Fill table:

x | f(x) calculation | f(x) value
----|--------------------------------------|----------
-2 | 2*(1/2)^{-2} + 3 = 2*(4) + 3 = 8+3=11 | 11
-1 | 2*(1/2)^{-1} + 3 = 2*(2) + 3 = 4+3=7 | 7
0 | 2*(1) + 3 = 2+3=5 | 5
1 | 2*(1/2) + 3 = 1 + 3 = 4 | 4
2 | 2*(1/4) + 3 = 0.5 + 3 = 3.5 | 3.5
3 | 2*(1/8) + 3 = 0.25 + 3 = 3.25 | 3.25

Characteristics:

- Growth or decay?
Base 1/2 < 1 → decay.

- Asymptote at?
As x→∞, (1/2)^x → 0, so f(x) → 3 → asymptote at y=3.

- Domain: All real numbers.

- Range: Starts high on left, decreases toward 3, never reaches → y > 3.

- y-intercept: x=0 → f(0)=5 → (0,5)

✔ Matches graph behavior: starts high, decreases, levels off above y=3.

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Final Answers:

1.
- Growth or decay? → Growth
- Asymptote at → y = 0
- Domain → All real numbers
- Range → y > 0
- y-intercept → (0, 2)

2.
- Growth or decay? → Decay
- Asymptote at → y = 4
- Domain → All real numbers
- Range → y > 4
- y-intercept → (0, 7)

3. Table filled as above; characteristics:
- Growth or decay? → Growth
- Asymptote at → y = -3
- Domain → All real numbers
- Range → y > -3
- y-intercept → (0, -2.75) or (0, -11/4)

4. Table filled as above; characteristics:
- Growth or decay? → Decay
- Asymptote at → y = 3
- Domain → All real numbers
- Range → y > 3
- y-intercept → (0, 5)

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Final Answer:

1. Growth; y = 0; all real numbers; y > 0; (0, 2)
2. Decay; y = 4; all real numbers; y > 4; (0, 7)
3. Growth; y = -3; all real numbers; y > -3; (0, -2.75)
4. Decay; y = 3; all real numbers; y > 3; (0, 5)