To graph the function \( y = 2 \left(\frac{1}{5}\right)^x \), we can start by calculating a couple of points.
Step 1: Calculate Points
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For \( x = 0 \): \[ y = 2 \left(\frac{1}{5}\right)^0 = 2 \times 1 = 2 \] So, the point is \( (0, 2) \).
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For \( x = 1 \): \[ y = 2 \left(\frac{1}{5}\right)^1 = 2 \times \frac{1}{5} = \frac{2}{5} = 0.4 \] So, the point is \( (1, 0.4) \).
You could choose a negative value for \( x \) as well:
- For \( x = -1 \): \[ y = 2 \left(\frac{1}{5}\right)^{-1} = 2 \times 5 = 10 \] So, the point is \( (-1, 10) \).
Step 2: Plot the Points
Now, we have the points:
- \( (0, 2) \)
- \( (1, 0.4) \)
- \( (-1, 10) \)
Step 3: Create the Graph
- The point \( (0, 2) \) indicates that the graph intercepts the y-axis at 2.
- The point \( (1, 0.4) \) shows that as \( x \) increases, \( y \) decreases since the function is an exponential decay.
- The point \( (-1, 10) \) shows that for negative \( x \), \( y \) increases significantly.
General Behavior of the Function
- The function \( y = 2 \left(\frac{1}{5}\right)^x \) approaches \( 0 \) as \( x \) increases towards positive infinity.
- As \( x \) decreases towards negative infinity, \( y \) increases without bound (towards infinity).
Graph Sketch
You can sketch the graph starting from the point \( (0, 2) \), moving towards \( (1, 0.4) \), and note the point \( (-1, 10) \) is higher on the graph. The curve should fall smoothly downwards as you move from left to right.
If you plot these points on the graph, you should see the characteristic shape of an exponential decay curve.