This worksheet guides students through analyzing exponential functions by identifying growth or decay, asymptotes, domain, range, and y-intercepts from graphs and tables.

Let’s solve each problem step by step, using the given functions and graphs (or tables) to find the required characteristics.

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## Problem 1:
Function: \( f(x) = 2(3)^x \)

• Growth or decay?

- The base is 3, which is greater than 1 → Exponential Growth

• Asymptote at ______

- Exponential functions of the form \( a(b)^x \) have a horizontal asymptote at y = 0 (the x-axis), unless shifted vertically.
- Here, no vertical shift → Asymptote at y = 0

• Domain:

- Exponential functions are defined for all real numbers → All real numbers or \( (-\infty, \infty) \)

• Range:

- Since it’s growth and starts above 0 (because coefficient 2 > 0), and approaches 0 as x → -∞, but never reaches it → y > 0 or \( (0, \infty) \)

• y-intercept:

- Set x = 0:
\( f(0) = 2(3)^0 = 2(1) = 2 \) → (0, 2)

✔ Answers for Problem 1:
- Growth
- Asymptote at y = 0
- Domain: All real numbers or \( (-\infty, \infty) \)
- Range: y > 0 or \( (0, \infty) \)
- y-intercept: (0, 2)

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## Problem 2:
Function: \( f(x) = 3\left(\frac{1}{2}\right)^x + 4 \)

• Growth or decay?

- Base is \( \frac{1}{2} \), which is between 0 and 1 → Exponential Decay

• Asymptote at ______

- The “+4” shifts the graph up 4 units.
- Original asymptote was y=0 → now it’s y = 4

• Domain:

- All real numbers → \( (-\infty, \infty) \)

• Range:

- Since it’s decay and shifted up 4, the function approaches 4 from above (never reaches it) → y > 4 or \( (4, \infty) \)

• y-intercept:

- Set x = 0:
\( f(0) = 3\left(\frac{1}{2}\right)^0 + 4 = 3(1) + 4 = 7 \) → (0, 7)

✔ Answers for Problem 2:
- Decay
- Asymptote at y = 4
- Domain: All real numbers or \( (-\infty, \infty) \)
- Range: y > 4 or \( (4, \infty) \)
- y-intercept: (0, 7)

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## Problem 3:
Function: \( f(x) = 4^{x-1} - 3 \)

First, fill in the table:

| x | f(x) = 4^{x-1} - 3 |
|----|---------------------|
| -2 | 4^{-3} - 3 = 1/64 - 3 = -2.984375 ≈ -2.98 |
| -1 | 4^{-2} - 3 = 1/16 - 3 = -2.9375 ≈ -2.94 |
| 0 | 4^{-1} - 3 = 1/4 - 3 = -2.75 |
| 1 | 4^{0} - 3 = 1 - 3 = -2 |
| 2 | 4^{1} - 3 = 4 - 3 = 1 |
| 3 | 4^{2} - 3 = 16 - 3 = 13 |

*(You can plot these points on the graph provided.)*

• Growth or decay?

- Base is 4 (>1) → Growth

• Asymptote at ______

- The “-3” shifts the graph down 3 units.
- Original asymptote y=0 → now y = -3

• Domain:

- All real numbers → \( (-\infty, \infty) \)

• Range:

- Approaches -3 from above → y > -3 or \( (-3, \infty) \)

• y-intercept:

- Set x = 0:
\( f(0) = 4^{-1} - 3 = \frac{1}{4} - 3 = -\frac{11}{4} = -2.75 \) → (0, -2.75)

✔ Answers for Problem 3:
- Growth
- Asymptote at y = -3
- Domain: All real numbers or \( (-\infty, \infty) \)
- Range: y > -3 or \( (-3, \infty) \)
- y-intercept: (0, -2.75)

*(Table filled as above)*

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## Problem 4:
Function: \( f(x) = 2\left(\frac{1}{2}\right)^x + 3 \)

Fill in the table:

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

*(Plot these points on the graph.)*

• Growth or decay?

- Base is \( \frac{1}{2} \) (<1) → Decay

• Asymptote at ______

- “+3” shifts graph up → y = 3

• Domain:

- All real numbers → \( (-\infty, \infty) \)

• Range:

- Approaches 3 from above → y > 3 or \( (3, \infty) \)

• y-intercept:

- Set x = 0:
\( f(0) = 2(1) + 3 = 5 \) → (0, 5)

✔ Answers for Problem 4:
- Decay
- Asymptote at y = 3
- Domain: All real numbers or \( (-\infty, \infty) \)
- Range: y > 3 or \( (3, \infty) \)
- y-intercept: (0, 5)

*(Table filled as above)*

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## ✔ Final Summary of Answers:

Problem 1:

- Growth
- Asymptote: y = 0
- Domain: All real numbers
- Range: y > 0
- y-intercept: (0, 2)

Problem 2:

- Decay
- Asymptote: y = 4
- Domain: All real numbers
- Range: y > 4
- y-intercept: (0, 7)

Problem 3:

- Growth
- Asymptote: y = -3
- Domain: All real numbers
- Range: y > -3
- y-intercept: (0, -2.75)
*(Table: x=-2→-2.98, x=-1→-2.94, x=0→-2.75, x=1→-2, x=2→1, x=3→13)*

Problem 4:

- Decay
- Asymptote: y = 3
- Domain: All real numbers
- Range: y > 3
- y-intercept: (0, 5)
*(Table: x=-2→11, x=-1→7, x=0→5, x=1→4, x=2→3.5, x=3→3.25)*

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Let me know if you’d like a sketch of any graph or further explanation!