To analyze and graph the function \( y = 2^{x + 3} - 1 \), let's follow the steps you provided:
Step 1: Find the asymptote
The function is an exponential function. The horizontal asymptote can be found by observing the behavior of the function as \( x \) approaches negative infinity.
As \( x \to -\infty \):
\[ y = 2^{x + 3} - 1 \to 0 - 1 = -1 \]
Thus, the equation of the asymptote is:
\[ y = -1 \]
Step 2: Plot at least four points with integer coordinates
Now we can choose various integer values for \( x \) to find the corresponding \( y \) values.
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For \( x = -3 \): \[ y = 2^{-3 + 3} - 1 = 2^0 - 1 = 1 - 1 = 0 \] Point: \((-3, 0)\)
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For \( x = -2 \): \[ y = 2^{-2 + 3} - 1 = 2^1 - 1 = 2 - 1 = 1 \] Point: \((-2, 1)\)
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For \( x = -1 \): \[ y = 2^{-1 + 3} - 1 = 2^2 - 1 = 4 - 1 = 3 \] Point: \((-1, 3)\)
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For \( x = 0 \): \[ y = 2^{0 + 3} - 1 = 2^3 - 1 = 8 - 1 = 7 \] Point: \((0, 7)\)
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For \( x = 1 \): \[ y = 2^{1 + 3} - 1 = 2^4 - 1 = 16 - 1 = 15 \] Point: \((1, 15)\)
Summary of Points
The points we calculated are:
- \((-3, 0)\)
- \((-2, 1)\)
- \((-1, 3)\)
- \((0, 7)\)
- \((1, 15)\)
Plotting the Graph
You can plot these points on a Cartesian coordinate system. The curve will approach the horizontal asymptote \(y = -1\) as \(x\) decreases and will rise steeply as \(x\) increases. Remember, the graph will shoot upwards as \(x\) increases beyond 0.
Final Note
You can use graphing software or a graphing calculator to visualize this function and ensure the points fall in line with your calculated values, as well as to see the asymptotic behavior.
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