Exponential Functions: Graphing worksheet featuring four functions to graph and analyze.

Problem Overview:

The task involves graphing four exponential functions using a table of values and defining their domains and ranges. The functions are:

1. \( y = 2^x \)
2. \( y = -(3^x) \)
3. \( y = \left(\frac{1}{2}\right)^x \)
4. \( y = \left(\frac{1}{4}\right)^{-x} \)

We will solve each part step by step.

---

1. \( y = 2^x \)

#### Step 1: Create a Table of Values
We calculate \( y = 2^x \) for the given \( x \)-values: \( -2, -1, 0, 1, 2 \).

\[
\begin{array}{c|c}
x & y = 2^x \\
\hline
-2 & 2^{-2} = \frac{1}{4} = 0.25 \\
-1 & 2^{-1} = \frac{1}{2} = 0.5 \\
0 & 2^0 = 1 \\
1 & 2^1 = 2 \\
2 & 2^2 = 4 \\
\end{array}
\]

#### Step 2: Graph the Function
Plot the points \((-2, 0.25)\), \((-1, 0.5)\), \((0, 1)\), \((1, 2)\), and \((2, 4)\) on the coordinate plane.

#### Step 3: Define the Domain and Range
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: All positive real numbers, \( (0, \infty) \).

---

2. \( y = -(3^x) \)

#### Step 1: Create a Table of Values
We calculate \( y = -(3^x) \) for the given \( x \)-values: \( -2, -1, 0, 1, 2 \).

\[
\begin{array}{c|c}
x & y = -(3^x) \\
\hline
-2 & -(3^{-2}) = -\frac{1}{9} \approx -0.111 \\
-1 & -(3^{-1}) = -\frac{1}{3} \approx -0.333 \\
0 & -(3^0) = -1 \\
1 & -(3^1) = -3 \\
2 & -(3^2) = -9 \\
\end{array}
\]

#### Step 2: Graph the Function
Plot the points \((-2, -0.111)\), \((-1, -0.333)\), \((0, -1)\), \((1, -3)\), and \((2, -9)\) on the coordinate plane.

#### Step 3: Define the Domain and Range
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: All negative real numbers, \( (-\infty, 0) \).

---

3. \( y = \left(\frac{1}{2}\right)^x \)

#### Step 1: Create a Table of Values
We calculate \( y = \left(\frac{1}{2}\right)^x \) for the given \( x \)-values: \( -2, -1, 0, 1, 2 \).

\[
\begin{array}{c|c}
x & y = \left(\frac{1}{2}\right)^x \\
\hline
-2 & \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 \\
-1 & \left(\frac{1}{2}\right)^{-1} = 2^1 = 2 \\
0 & \left(\frac{1}{2}\right)^0 = 1 \\
1 & \left(\frac{1}{2}\right)^1 = \frac{1}{2} = 0.5 \\
2 & \left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25 \\
\end{array}
\]

#### Step 2: Graph the Function
Plot the points \((-2, 4)\), \((-1, 2)\), \((0, 1)\), \((1, 0.5)\), and \((2, 0.25)\) on the coordinate plane.

#### Step 3: Define the Domain and Range
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: All positive real numbers, \( (0, \infty) \).

---

4. \( y = \left(\frac{1}{4}\right)^{-x} \)

#### Step 1: Simplify the Function
Using the property of exponents \( a^{-b} = \frac{1}{a^b} \), we can rewrite the function as:
\[
y = \left(\frac{1}{4}\right)^{-x} = 4^x
\]

#### Step 2: Create a Table of Values
We calculate \( y = 4^x \) for the given \( x \)-values: \( -2, -1, 0, 1, 2 \).

\[
\begin{array}{c|c}
x & y = 4^x \\
\hline
-2 & 4^{-2} = \frac{1}{4^2} = \frac{1}{16} = 0.0625 \\
-1 & 4^{-1} = \frac{1}{4} = 0.25 \\
0 & 4^0 = 1 \\
1 & 4^1 = 4 \\
2 & 4^2 = 16 \\
\end{array}
\]

#### Step 3: Graph the Function
Plot the points \((-2, 0.0625)\), \((-1, 0.25)\), \((0, 1)\), \((1, 4)\), and \((2, 16)\) on the coordinate plane.

#### Step 4: Define the Domain and Range
- Domain: All real numbers, \( (-\infty, \infty) \).
- Range: All positive real numbers, \( (0, \infty) \).

---

Final Answers:

1. \( y = 2^x \):
- Domain: \( (-\infty, \infty) \)
- Range: \( (0, \infty) \)

2. \( y = -(3^x) \):
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, 0) \)

3. \( y = \left(\frac{1}{2}\right)^x \):
- Domain: \( (-\infty, \infty) \)
- Range: \( (0, \infty) \)

4. \( y = \left(\frac{1}{4}\right)^{-x} \):
- Domain: \( (-\infty, \infty) \)
- Range: \( (0, \infty) \)

\[
\boxed{
\begin{array}{l}
1. \text{Domain: } (-\infty, \infty), \text{ Range: } (0, \infty) \\
2. \text{Domain: } (-\infty, \infty), \text{ Range: } (-\infty, 0) \\
3. \text{Domain: } (-\infty, \infty), \text{ Range: } (0, \infty) \\
4. \text{Domain: } (-\infty, \infty), \text{ Range: } (0, \infty) \\
\end{array}
}
\]