Question

Jeremy claims that if a linear function has a slope of the same steepness and the same y-intercept as the linear function in the graph, then it must be the same function.

On a coordinate plane, a line goes through points (0, negative 1) and (2, 0).

Which equation proves Jeremy’s argument is incorrect?
y = negative one-half x minus 1
y = negative one-half x + 1
y = one-half x minus 1
y = one-half x + 1

Answer 1

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).

1. **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):
\[
m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2}
\]

2. **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:

1. \(y = - \frac{1}{2}x - 1\) has slope \(-\frac{1}{2}\) and y-intercept \(-1\).
2. \(y = - \frac{1}{2}x + 1\) has slope \(-\frac{1}{2}\) and y-intercept \(1\).
3. \(y = \frac{1}{2}x - 1\) has slope \(\frac{1}{2}\) and y-intercept \(-1\) (This is the same as our derived line).
4. \(y = \frac{1}{2}x + 1\) 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**.