To analyze and graph the function \( y = \left( \frac{1}{2} \right)^{x+2} - 9 \), we can start by identifying the key features of the function, such as its asymptote and points for plotting.
Step 1: Determine the Asymptote
The asymptote of exponential functions is generally determined by the constant term in the equation. Here, the function approaches the value of \(-9\) as \(x\) approaches infinity. Therefore, the equation of the asymptote is: \[ y = -9 \]
Step 2: Find Points with Integer Coordinates
To find points on the graph, we'll calculate the value of \(y\) for a range of \(x\) values (let's choose \(x = -5\) to \(x = 3\)):
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For \(x = -5\): \[ y = \left( \frac{1}{2} \right)^{-5+2} - 9 = \left( \frac{1}{2} \right)^{-3} - 9 = 2^3 - 9 = 8 - 9 = -1 \] So the point is \((-5, -1)\).
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For \(x = -4\): \[ y = \left( \frac{1}{2} \right)^{-4+2} - 9 = \left( \frac{1}{2} \right)^{-2} - 9 = 2^2 - 9 = 4 - 9 = -5 \] So the point is \((-4, -5)\).
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For \(x = -3\): \[ y = \left( \frac{1}{2} \right)^{-3+2} - 9 = \left( \frac{1}{2} \right)^{-1} - 9 = 2^1 - 9 = 2 - 9 = -7 \] So the point is \((-3, -7)\).
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For \(x = -2\): \[ y = \left( \frac{1}{2} \right)^{-2+2} - 9 = \left( \frac{1}{2} \right)^{0} - 9 = 1 - 9 = -8 \] So the point is \((-2, -8)\).
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For \(x = -1\): \[ y = \left( \frac{1}{2} \right)^{-1+2} - 9 = \left( \frac{1}{2} \right)^{1} - 9 = \frac{1}{2} - 9 = -8.5 \] (not an integer point)
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For \(x = 0\): \[ y = \left( \frac{1}{2} \right)^{0+2} - 9 = \left( \frac{1}{2} \right)^{2} - 9 = \frac{1}{4} - 9 = -8.75 \] (not an integer point)
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For \(x = 1\): \[ y = \left( \frac{1}{2} \right)^{1+2} - 9 = \left( \frac{1}{2} \right)^{3} - 9 = \frac{1}{8} - 9 = -8.875 \] (not an integer point)
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For \(x = 2\): \[ y = \left( \frac{1}{2} \right)^{2+2} - 9 = \left( \frac{1}{2} \right)^{4} - 9 = \frac{1}{16} - 9 = -8.9375 \] (not an integer point)
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For \(x = 3\): \[ y = \left( \frac{1}{2} \right)^{3+2} - 9 = \left( \frac{1}{2} \right)^{5} - 9 = \frac{1}{32} - 9 = -8.96875 \] (not an integer point)
Summary of Points with Integer Coordinates
- \((-5, -1)\)
- \((-4, -5)\)
- \((-3, -7)\)
- \((-2, -8)\)
Graphing
You would plot these points on a graph, drawing the horizontal line \(y = -9\) for the asymptote. As \(x\) increases, the function descends toward the asymptote without ever crossing or touching it.
Conclusion
These procedures allow us to analyze and graph the function \(y = \left( \frac{1}{2} \right)^{x+2} - 9\) while identifying key features like the asymptote and specific coordinates for plotting points. The points can be represented on a coordinate system for visualization.
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