To analyze the function \( y = \left( \frac{1}{2} \right)^{x + 2} - 9 \), let's find the horizontal asymptote and plot points.
Step 1: Find the Equation of the Asymptote
As \( x \) approaches infinity, \( \left( \frac{1}{2} \right)^{x + 2} \) approaches 0. Therefore, the horizontal asymptote for the function is:
\[ y = -9 \]
Step 2: Calculate Points
Let's substitute five integer values for \( x \) to calculate \( y \):
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For \( x = -5 \): \[ y = \left( \frac{1}{2} \right)^{-5 + 2} - 9 = \left( \frac{1}{2} \right)^{-3} - 9 = 8 - 9 = -1 \] Point: \((-5, -1)\)
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For \( x = -4 \): \[ y = \left( \frac{1}{2} \right)^{-4 + 2} - 9 = \left( \frac{1}{2} \right)^{-2} - 9 = 4 - 9 = -5 \] Point: \((-4, -5)\)
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For \( x = -3 \): \[ y = \left( \frac{1}{2} \right)^{-3 + 2} - 9 = \left( \frac{1}{2} \right)^{-1} - 9 = 2 - 9 = -7 \] Point: \((-3, -7)\)
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For \( x = -2 \): \[ y = \left( \frac{1}{2} \right)^{-2 + 2} - 9 = \left( \frac{1}{2} \right)^{0} - 9 = 1 - 9 = -8 \] Point: \((-2, -8)\)
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For \( x = -1 \): \[ y = \left( \frac{1}{2} \right)^{-1 + 2} - 9 = \left( \frac{1}{2} \right)^{1} - 9 = \frac{1}{2} - 9 = -\frac{17}{2} \approx -8.5 \] Point: \((-1, -8.5)\)
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For \( x = 0 \): \[ y = \left( \frac{1}{2} \right)^{0 + 2} - 9 = \left( \frac{1}{2} \right)^{2} - 9 = \frac{1}{4} - 9 = -\frac{35}{4} \approx -8.75 \] Point: \((0, -8.75)\)
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For \( x = 1 \): \[ y = \left( \frac{1}{2} \right)^{1 + 2} - 9 = \left( \frac{1}{2} \right)^{3} - 9 = \frac{1}{8} - 9 = -\frac{71}{8} \approx -8.875 \] Point: \((1, -8.875)\)
Step 3: Summary of Points
Here’s the list of points we obtained:
- \((-5, -1)\)
- \((-4, -5)\)
- \((-3, -7)\)
- \((-2, -8)\)
- \((-1, -8.5)\)
- \( (0, -8.75) \)
- \( (1, -8.875) \)
Step 4: Graphing the Function
To graph the points:
- Start by plotting the asymptote line at \( y = -9 \).
- Then plot each of the calculated points on a coordinate grid.
- Connect the points smoothly to show the curve approaching the asymptote as \( x \) increases.
Here's a rough description of how it will look:
- The graph will start above the asymptote and drop towards it as \( x \) increases, flattening out near \( y = -9 \).
Consider using graphing software or a calculator to visualize this. If you'd like to graph it manually, just ensure that your points are correctly plotted and that you draw the curve approaching the horizontal asymptote at \( y = -9 \).