To graph the function \( y = \frac{3}{7^x} + 28 \), follow these steps:
Step 1: Understand the basic function
The function can be analyzed piece by piece:
- The term \( \frac{3}{7^x} \) is a decreasing function that approaches 0 as \( x \) increases.
- Adding 28 shifts the entire graph up by 28 units.
Step 2: Determine key points
Here are some key points to evaluate the function:
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When \( x = 0 \): \[ y = \frac{3}{7^0} + 28 = \frac{3}{1} + 28 = 31 \]
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When \( x = 1 \): \[ y = \frac{3}{7^1} + 28 = \frac{3}{7} + 28 \approx 28.43 \]
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When \( x = 2 \): \[ y = \frac{3}{7^2} + 28 = \frac{3}{49} + 28 \approx 28.06 \]
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When \( x = 3 \): \[ y = \frac{3}{7^3} + 28 = \frac{3}{343} + 28 \approx 28.01 \]
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When \( x = 4 \): \[ y = \frac{3}{7^4} + 28 = \frac{3}{2401} + 28 \approx 28.00 \]
Step 3: Identify the asymptote
As \( x \) approaches infinity, \( \frac{3}{7^x} \) approaches 0, so:
\[ y \rightarrow 28 \]
Thus, \( y = 28 \) is a horizontal asymptote.
Step 4: Setup the axes
- X-axis from 0 to 8.
- Y-axis from 0 to 32.
Step 5: Draw the graph
Plot the points calculated above:
- At \( x = 0, y = 31 \).
- At \( x = 1, y \approx 28.43 \).
- At \( x = 2, y \approx 28.06 \).
- At \( x = 3, y \approx 28.01 \).
- At \( x = 4, y \approx 28.00 \).
Final Graph
You would see:
- The curve starts at \( (0, 31) \) and gradually approaches the horizontal line \( y = 28 \) as \( x \) increases toward 8.
- Lay out the graph in the defined axes with the curve visibly decreasing and leveling off near \( y = 28 \).
Note
If you have graphing software or tools, you could plot these functions to see the behavior more clearly. If you are drawing manually, use smooth lines to connect points, approaching the asymptote.
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