Draw the graphs `y=2^(x)` and `y=((1)/(2))^(x)`, on the same diagram, for `-3lexle3`

Answer

To draw the graphs of \( y = 2^x \) and \( y = \left(\frac{1}{2}\right)^x \) for \( -3 \leq x \leq 3 \), we will follow these steps:

### Step 1: Create a table of values for \( y = 2^x \)

We will calculate the values of \( y \) for various values of \( x \) in the range from -3 to 3.

- For \( x = -3 \):
\[ y = 2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125 \]

- For \( x = -2 \):
\[ y = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25 \]

- For \( x = -1 \):
\[ y = 2^{-1} = \frac{1}{2} = 0.5 \]

- For \( x = 0 \):
\[ y = 2^0 = 1 \]

- For \( x = 1 \):
\[ y = 2^1 = 2 \]

- For \( x = 2 \):
\[ y = 2^2 = 4 \]

- For \( x = 3 \):
\[ y = 2^3 = 8 \]

**Table for \( y = 2^x \)**:
\[ \begin{array}{|c|c|} \hline x & y = 2^x \\ \hline -3 & 0.125 \\ -2 & 0.25 \\ -1 & 0.5 \\ 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ 3 & 8 \\ \hline \end{array} \]

### Step 2: Create a table of values for \( y = \left(\frac{1}{2}\right)^x \)

Now we will calculate the values of \( y \) for the same values of \( x \).

- For \( x = -3 \):
\[ y = \left(\frac{1}{2}\right)^{-3} = 2^3 = 8 \]

- For \( x = -2 \):
\[ y = \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 \]

- For \( x = -1 \):
\[ y = \left(\frac{1}{2}\right)^{-1} = 2^1 = 2 \]

- For \( x = 0 \):
\[ y = \left(\frac{1}{2}\right)^{0} = 1 \]

- For \( x = 1 \):
\[ y = \left(\frac{1}{2}\right)^{1} = 0.5 \]

- For \( x = 2 \):
\[ y = \left(\frac{1}{2}\right)^{2} = 0.25 \]

- For \( x = 3 \):
\[ y = \left(\frac{1}{2}\right)^{3} = 0.125 \]

**Table for \( y = \left(\frac{1}{2}\right)^x \)**:
\[ \begin{array}{|c|c|} \hline x & y = \left(\frac{1}{2}\right)^x \\ \hline -3 & 8 \\ -2 & 4 \\ -1 & 2 \\ 0 & 1 \\ 1 & 0.5 \\ 2 & 0.25 \\ 3 & 0.125 \\ \hline \end{array} \]

### Step 3: Plot the points on the graph

Using the values from the tables, we can plot the points for both functions on the same graph.

- For \( y = 2^x \):
- Points: \( (-3, 0.125), (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8) \)

- For \( y = \left(\frac{1}{2}\right)^x \):
- Points: \( (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 0.5), (2, 0.25), (3, 0.125) \)

### Step 4: Draw the graphs

1. Draw the axes (x-axis and y-axis).
2. Plot the points for \( y = 2^x \) in one color (e.g., blue).
3. Plot the points for \( y = \left(\frac{1}{2}\right)^x \) in another color (e.g., red).
4. Connect the points smoothly to form the curves of the graphs.