To check Jeremy's claim, we first need to determine the slope and y-intercept of the line that passes through points (0, -1) and (2, 0).
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Finding the slope (m): The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using points (0, -1) and (2, 0): \[ m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \]
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Finding the y-intercept (b): The y-intercept (b) is clearly given by the point (0, -1), so: \[ b = -1 \]
Thus, the equation of the line in slope-intercept form \(y = mx + b\) is: \[ y = \frac{1}{2}x - 1 \]
Next, we compare this with the options provided. Jeremy claims that if two linear functions have the same slope (steepness) and the same y-intercept, then they must be the same function.
Now, let's analyze the given options for their slopes and y-intercepts:
- \(y = - \frac{1}{2}x - 1\) has slope \(-\frac{1}{2}\) and y-intercept \(-1\).
- \(y = - \frac{1}{2}x + 1\) has slope \(-\frac{1}{2}\) and y-intercept \(1\).
- \(y = \frac{1}{2}x - 1\) has slope \(\frac{1}{2}\) and y-intercept \(-1\) (This is the same as our derived line).
- \(y = \frac{1}{2}x + 1\) has slope \(\frac{1}{2}\) and y-intercept \(1\).
Now, we look for an equation that has the same slope and y-intercept:
- The equation that matches the slope of \(\frac{1}{2}\) and y-intercept of \(-1\) is: \[ y = \frac{1}{2}x - 1 \]
Thus, to disprove Jeremy's argument, we take the other option with the same slope but different y-intercept:
The equation that serves as a counter-example to Jeremy's claim is: \[ y = \frac{1}{2}x + 1 \]
This has the same slope as our line but a different y-intercept, proving that it is not the same function.
Thus, the correct response to prove Jeremy's argument incorrect is: y = \frac{1}{2}x + 1.
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