To analyze and graph the function \( y = 2^{(x - 2)} - 5 \), let's begin with the following steps:
Step 1: Determine the Asymptote
The function \( y = 2^{(x - 2)} - 5 \) is an exponential function that is shifted down by 5 units. The horizontal asymptote can be determined by considering the limit of the function as \( x \) approaches negative infinity:
\[ \lim_{x \to -\infty} (2^{(x - 2)} - 5) = 0 - 5 = -5 \]
So, the equation of the asymptote is: \[ y = -5 \]
Step 2: Find Points on the Graph
To plot the function, we can substitute various integer values of \( x \) to find corresponding \( y \) values. Below are calculations for several integer values of \( x \):
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For \( x = 0 \): \[ y = 2^{(0 - 2)} - 5 = 2^{-2} - 5 = \frac{1}{4} - 5 = -4.75 \]
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For \( x = 1 \): \[ y = 2^{(1 - 2)} - 5 = 2^{-1} - 5 = \frac{1}{2} - 5 = -4.5 \]
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For \( x = 2 \): \[ y = 2^{(2 - 2)} - 5 = 2^{0} - 5 = 1 - 5 = -4 \]
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For \( x = 3 \): \[ y = 2^{(3 - 2)} - 5 = 2^{1} - 5 = 2 - 5 = -3 \]
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For \( x = 4 \): \[ y = 2^{(4 - 2)} - 5 = 2^{2} - 5 = 4 - 5 = -1 \]
Summary of Points
Based on these calculations, we have the following integer coordinate points:
- \( (0, -4.75) \)
- \( (1, -4.5) \)
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)
Step 3: Create a Graph
To graph this function, plot the points on the Cartesian coordinate system and draw the horizontal asymptote \( y = -5 \).
Points Summary for Graphing
Here are the points you can plot on a coordinate grid, rounding the y-values where necessary:
- \( (0, -5) \) (Approx)
- \( (1, -5) \) (Approx)
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)
You can plot these points, and connect them smoothly to illustrate the exponential shape, approaching the asymptote \( y = -5 \) as \( x \) decreases.
Your final output for the graph would include these points plotted on the axes below, with a dashed line for the asymptote at \( y = -5 \).
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