To analyze the function \( y = 2^{(x - 2)} - 5 \) and find its features including the asymptote and integer coordinate points, we can follow these steps:
Step 1: Determine the Equation of Asymptote
The function \( y = 2^{(x - 2)} - 5 \) is an exponential function. The horizontal asymptote occurs as \( x \to \infty \) or \( x \to -\infty \).
To find the asymptote, we can observe that as \( x \to -\infty \), \( 2^{(x - 2)} \) approaches 0. Therefore, the asymptote is:
\[ y = -5 \]
Step 2: Calculate Points with Integer Coordinates
Now we can calculate points by choosing integer values of \( x \):
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For \( x = 0 \): \[ y = 2^{(0 - 2)} - 5 = 2^{-2} - 5 = \frac{1}{4} - 5 = -\frac{19}{4} \approx -4.75 \]
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For \( x = 1 \): \[ y = 2^{(1 - 2)} - 5 = 2^{-1} - 5 = \frac{1}{2} - 5 = -\frac{9}{2} = -4.5 \]
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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 \( x = 4 \): \[ y = 2^{(4 - 2)} - 5 = 2^{2} - 5 = 4 - 5 = -1 \]
The integer coordinate points we can plot are:
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)
Summary of Calculated Points
The integer coordinate points are:
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)
Step 3: Graph the Function
Now, we will plot the points and include the asymptote at \( y = -5 \).
Graph Overview:
- The asymptote line: \( y = -5 \)
- Points to plot:
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)
Graphing: You can graph this function using graphing software or a graphing calculator to visualize how the function approaches the asymptote \( y = -5 \) and includes the points listed above.
The graph would show the exponential growth on the right side, starting from the asymptote on the left and passing through the calculated points.
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