Question

Graph the function y = 3 * 1/2 to the power of x

Answer 1

To graph the function \( y = 3 \left(\frac{1}{2}\right)^x \), we can follow these steps:

### Step 1: Identify Key Features of the Function
1. **Y-Intercept**: Set \( x = 0 \).
\[
y = 3 \left(\frac{1}{2}\right)^0 = 3 \times 1 = 3
\]
So, the y-intercept is at the point (0, 3).

2. **Behavior as \( x \to \infty \)**: As \( x \) increases, \( \left(\frac{1}{2}\right)^x \) approaches 0. Thus, \( y \) approaches 0 but never touches it.

3. **Behavior as \( x \to -\infty \)**: As \( x \) decreases, \( \left(\frac{1}{2}\right)^x \) increases. Specifically, \( y \) approaches infinity for very large negative values of \( x \).

### Step 2: Calculate Additional Points
To get a good idea of the function's shape, calculate a few additional points:

- For \( x = 1 \):
\[
y = 3 \left(\frac{1}{2}\right)^1 = 3 \times \frac{1}{2} = 1.5
\]
Point: (1, 1.5)

- For \( x = 2 \):
\[
y = 3 \left(\frac{1}{2}\right)^2 = 3 \times \frac{1}{4} = 0.75
\]
Point: (2, 0.75)

- For \( x = -1 \):
\[
y = 3 \left(\frac{1}{2}\right)^{-1} = 3 \times 2 = 6
\]
Point: (-1, 6)

- For \( x = -2 \):
\[
y = 3 \left(\frac{1}{2}\right)^{-2} = 3 \times 4 = 12
\]
Point: (-2, 12)

### Step 3: Plot the Points
Now we can plot these points on a graph:
- (0, 3)
- (1, 1.5)
- (2, 0.75)
- (-1, 6)
- (-2, 12)

### Step 4: Draw the Curve
- Start the curve from the point (-2, 12), which moves downward through (-1, 6), crossing the y-axis at (0, 3), and then approaching (1, 1.5), and (2, 0.75).
- As \( x \) increases, the curve approaches the x-axis (y = 0) but never actually touches it.

### Final Shape
The function \( y = 3 \left(\frac{1}{2}\right)^x \) creates a decreasing exponential curve starting high when \( x \) is negative and approaching zero as \( x \) becomes positive.

You should end up with a graph that looks like this:

```
y
|
| *
| *
| *
| *
| *
| *
|*
|___________________ x
```

### Asymptote
The x-axis (y = 0) is a horizontal asymptote for this function.

Make sure to label your axes and provide a scale if you are drawing this graph on paper or in a digital tool!