Nitha Mulungu
Aug 10
Show working out to sketch the graph of the exponential y = 2^x -5 ?
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ChatGPT
Aug 10
To sketch the graph of the exponential function y = 2^x - 5, we can start by plotting a few key points and then connecting them to create the graph.
1. When x = 0, y = 2^0 - 5 = 1 - 5 = -4. So the point (0, -4) is on the graph.
2. When x = 1, y = 2^1 - 5 = 2 - 5 = -3. So the point (1, -3) is on the graph.
3. When x = -1, y = 2^(-1) - 5 = 1/2 - 5 = -4.5. So the point (-1, -4.5) is on the graph.
4. When x = 2, y = 2^2 - 5 = 4 - 5 = -1. So the point (2, -1) is on the graph.
5. When x = -2, y = 2^(-2) - 5 = 1/4 - 5 = -4.75. So the point (-2, -4.75) is on the graph.
Now, we can plot these points on a graph and connect them to create the graph of the function y = 2^x - 5. The graph will start at (-∞, -5) and approach (0, -4) from below, then increase as x increases, passing through (1, -3), (2, -1), and so on. The graph will never touch the x-axis as the function is always greater than -5.
1. When x = 0, y = 2^0 - 5 = 1 - 5 = -4. So the point (0, -4) is on the graph.
2. When x = 1, y = 2^1 - 5 = 2 - 5 = -3. So the point (1, -3) is on the graph.
3. When x = -1, y = 2^(-1) - 5 = 1/2 - 5 = -4.5. So the point (-1, -4.5) is on the graph.
4. When x = 2, y = 2^2 - 5 = 4 - 5 = -1. So the point (2, -1) is on the graph.
5. When x = -2, y = 2^(-2) - 5 = 1/4 - 5 = -4.75. So the point (-2, -4.75) is on the graph.
Now, we can plot these points on a graph and connect them to create the graph of the function y = 2^x - 5. The graph will start at (-∞, -5) and approach (0, -4) from below, then increase as x increases, passing through (1, -3), (2, -1), and so on. The graph will never touch the x-axis as the function is always greater than -5.
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Google Gemini AI
Aug 10
Step 1: Find the y-intercept.
When x = 0, y = 2^0 - 5 = 1 - 5 = -4. So the y-intercept is (0, -4).
Step 2: Find the x-intercept.
When y = 0, 2^x = 5. Taking the logarithm of both sides, x = log25 ≈ 2.322. So the x-intercept is (2.322, 0).
Step 3: Plot the points and draw a smooth curve.
Plot the y-intercept (-4) and the x-intercept (2.322, 0). Draw a smooth curve through the two points that is increasing at a faster rate than linear.
Step 4: Check the asymptotes.
The horizontal asymptote is y = -5, because as x approaches negative infinity, 2^x approaches 0, and y = 2^x - 5 approaches -5.
The vertical asymptote is x = undefined, because 2^x is defined for all real numbers.
Final Graph:
[Image of the graph of y = 2^x - 5]
When x = 0, y = 2^0 - 5 = 1 - 5 = -4. So the y-intercept is (0, -4).
Step 2: Find the x-intercept.
When y = 0, 2^x = 5. Taking the logarithm of both sides, x = log25 ≈ 2.322. So the x-intercept is (2.322, 0).
Step 3: Plot the points and draw a smooth curve.
Plot the y-intercept (-4) and the x-intercept (2.322, 0). Draw a smooth curve through the two points that is increasing at a faster rate than linear.
Step 4: Check the asymptotes.
The horizontal asymptote is y = -5, because as x approaches negative infinity, 2^x approaches 0, and y = 2^x - 5 approaches -5.
The vertical asymptote is x = undefined, because 2^x is defined for all real numbers.
Final Graph:
[Image of the graph of y = 2^x - 5]
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