To graph the function \( y = 3 \left( \frac{1}{12} \right)^x \), we can plot a couple of points by selecting values for \( x \) and calculating the corresponding \( y \) values.
Step 1: Choose values for \( x \)
Let's choose two values for \( x \):
- \( x = 0 \)
- \( x = 1 \)
Step 2: Calculate corresponding \( y \) values
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For \( x = 0 \): \[ y = 3 \left( \frac{1}{12} \right)^0 = 3 \cdot 1 = 3 \] So, one point is \( (0, 3) \).
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For \( x = 1 \): \[ y = 3 \left( \frac{1}{12} \right)^1 = 3 \cdot \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \] So, another point is \( (1, \frac{1}{4}) \).
Step 3: Plot the points
The points we have are:
- \( (0, 3) \)
- \( (1, \frac{1}{4}) \)
Step 4: Describe the graph
To sketch the graph:
- The point \( (0, 3) \) indicates that when \( x = 0 \), the function starts at \( y = 3 \).
- The point \( (1, \frac{1}{4}) \) shows that the function decreases rapidly as \( x \) increases since \( \frac{1}{4} \) is much less than \( 3 \).
Since the base \( \frac{1}{12} \) is less than 1, the function will exhibit exponential decay, approaching \( y = 0 \) as \( x \) increases, but never actually touching the x-axis.
Visualization
Here's a simple text diagram representation:
y
|
3 | *
|
|
|
1 | |
|
|
0 +-------|-------|-------|----- x
0 1 2 3
The asterisk (*) represents the point \( (0, 3) \), and the vertical location of \( (1, \frac{1}{4}) \) would be slightly above 0 but significantly lower than \( y = 3 \).
If you were using graphing software or graph paper, you would see the curve approaching zero as \( x \) increases.