This worksheet helps students master exponential functions by analyzing graphs and completing tables to identify key characteristics.

Problem Analysis and Solution

The worksheet involves analyzing exponential functions using graphs and tables. Let's solve each problem step by step.

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

- Graph Characteristics:
- The graph shows an exponential growth function because it increases rapidly as \( x \) increases.
- The asymptote is at \( y = 0 \) (the x-axis), as the function approaches but never touches this line.
- The domain is all real numbers, \( (-\infty, \infty) \).
- The range is \( (0, \infty) \) because the function is always positive.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 2(3)^0 = 2(1) = 2
\]
So, the y-intercept is \( (0, 2) \).

Answers:
- Growth or decay? Growth
- Asymptote at: \( y = 0 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (0, \infty) \)
- y-intercept: \( (0, 2) \)

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

- Graph Characteristics:
- The graph shows an exponential decay function because it decreases as \( x \) increases.
- The asymptote is at \( y = 4 \), as the function approaches but never touches this line.
- The domain is all real numbers, \( (-\infty, \infty) \).
- The range is \( (4, \infty) \) because the function is always greater than 4.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 3\left(\frac{1}{2}\right)^0 + 4 = 3(1) + 4 = 7
\]
So, the y-intercept is \( (0, 7) \).

Answers:
- Growth or decay? Decay
- Asymptote at: \( y = 4 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (4, \infty) \)
- y-intercept: \( (0, 7) \)

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

- Step 1: Create the x-y table.
\[
\begin{array}{c|c}
x & f(x) = 4^{x-1} - 3 \\
\hline
-2 & 4^{-2-1} - 3 = 4^{-3} - 3 = \frac{1}{64} - 3 = -2.984375 \\
-1 & 4^{-1-1} - 3 = 4^{-2} - 3 = \frac{1}{16} - 3 = -2.9375 \\
0 & 4^{0-1} - 3 = 4^{-1} - 3 = \frac{1}{4} - 3 = -2.75 \\
1 & 4^{1-1} - 3 = 4^0 - 3 = 1 - 3 = -2 \\
2 & 4^{2-1} - 3 = 4^1 - 3 = 4 - 3 = 1 \\
3 & 4^{3-1} - 3 = 4^2 - 3 = 16 - 3 = 13 \\
\end{array}
\]

- Graph Characteristics:
- The graph shows exponential growth because the function increases as \( x \) increases.
- The asymptote is at \( y = -3 \), as the function approaches but never touches this line.
- The domain is all real numbers, \( (-\infty, \infty) \).
- The range is \( (-3, \infty) \) because the function is always greater than -3.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 4^{0-1} - 3 = 4^{-1} - 3 = \frac{1}{4} - 3 = -2.75
\]
So, the y-intercept is \( (0, -2.75) \).

Answers:
- Growth or decay? Growth
- Asymptote at: \( y = -3 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (-3, \infty) \)
- y-intercept: \( (0, -2.75) \)

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

- Step 1: Create the x-y table.
\[
\begin{array}{c|c}
x & f(x) = 2\left(\frac{1}{2}\right)^x + 3 \\
\hline
-2 & 2\left(\frac{1}{2}\right)^{-2} + 3 = 2(4) + 3 = 8 + 3 = 11 \\
-1 & 2\left(\frac{1}{2}\right)^{-1} + 3 = 2(2) + 3 = 4 + 3 = 7 \\
0 & 2\left(\frac{1}{2}\right)^0 + 3 = 2(1) + 3 = 2 + 3 = 5 \\
1 & 2\left(\frac{1}{2}\right)^1 + 3 = 2\left(\frac{1}{2}\right) + 3 = 1 + 3 = 4 \\
2 & 2\left(\frac{1}{2}\right)^2 + 3 = 2\left(\frac{1}{4}\right) + 3 = \frac{1}{2} + 3 = 3.5 \\
3 & 2\left(\frac{1}{2}\right)^3 + 3 = 2\left(\frac{1}{8}\right) + 3 = \frac{1}{4} + 3 = 3.25 \\
\end{array}
\]

- Graph Characteristics:
- The graph shows exponential decay because the function decreases as \( x \) increases.
- The asymptote is at \( y = 3 \), as the function approaches but never touches this line.
- The domain is all real numbers, \( (-\infty, \infty) \).
- The range is \( (3, \infty) \) because the function is always greater than 3.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = 2\left(\frac{1}{2}\right)^0 + 3 = 2(1) + 3 = 2 + 3 = 5
\]
So, the y-intercept is \( (0, 5) \).

Answers:
- Growth or decay? Decay
- Asymptote at: \( y = 3 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (3, \infty) \)
- y-intercept: \( (0, 5) \)

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Final Answers:

1.
- Growth or decay? Growth
- Asymptote at: \( y = 0 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (0, \infty) \)
- y-intercept: \( (0, 2) \)

2.
- Growth or decay? Decay
- Asymptote at: \( y = 4 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (4, \infty) \)
- y-intercept: \( (0, 7) \)

3.
- Growth or decay? Growth
- Asymptote at: \( y = -3 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (-3, \infty) \)
- y-intercept: \( (0, -2.75) \)

4.
- Growth or decay? Decay
- Asymptote at: \( y = 3 \)
- Domain: \( (-\infty, \infty) \)
- Range: \( (3, \infty) \)
- y-intercept: \( (0, 5) \)

\boxed{
\begin{array}{l}
\text{1. Growth, } y = 0, (-\infty, \infty), (0, \infty), (0, 2) \\
\text{2. Decay, } y = 4, (-\infty, \infty), (4, \infty), (0, 7) \\
\text{3. Growth, } y = -3, (-\infty, \infty), (-3, \infty), (0, -2.75) \\
\text{4. Decay, } y = 3, (-\infty, \infty), (3, \infty), (0, 5)
\end{array}
}