Question

the graph shows the relationship between x, the number of lawns brandon cuts, and y, his net profit in tens of dollars, after printing flyers for his business. which equation matches the graph? show your work here \\(\boldsymbol{y = \frac{3}{2}x + 1}\\) \\(\boldsymbol{y = \frac{1}{3}x + 1}\\) \\(\boldsymbol{y = \frac{1}{2}x - 1}\\) \\(\boldsymbol{y = \frac{3}{2}x - 1}\\)

Explanation:

Step1: Identify the y-intercept

The graph passes through (0, -1), so the y-intercept \( b = -1 \). This eliminates options with \( b = 1 \).

Step2: Calculate the slope

Use two points, e.g., (0, -1) and (2, 0). Slope \( m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \)? Wait, no, wait. Wait, looking at the graph, when x=2, y=0? Wait, no, let's check again. Wait, the graph starts at (0, -1) and goes up. Let's take (2, 0)? Wait, no, maybe (4, 1)? Wait, no, let's recalculate. Wait, the line passes through (0, -1) and (2, 0)? Wait, no, when x=2, y=0? Wait, no, let's check the grid. The y-axis: the bottom is -1? Wait, the graph has a point at (0, -1) (since it starts below the origin) and then when x=2, y=0? Wait, no, maybe (2, 0) is a point? Wait, no, let's use (0, -1) and (2, 0). Then slope \( m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \)? Wait, but the options have \( \frac{1}{2}x - 1 \) and \( \frac{3}{2}x - 1 \). Wait, maybe another point. Let's take (2, 0) and (4, 1)? No, wait, maybe (0, -1) and (2, 0): slope \( \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \). Wait, but the option \( y = \frac{1}{2}x - 1 \) has slope \( \frac{1}{2} \) and y-intercept -1. Wait, but let's check another point. If x=4, then \( y = \frac{1}{2}(4) - 1 = 2 - 1 = 1 \). Does the graph pass through (4, 1)? Yes, that seems right. Wait, but wait, maybe I made a mistake. Wait, the other option with slope \( \frac{3}{2} \): if x=2, \( y = \frac{3}{2}(2) - 1 = 3 - 1 = 2 \), but the graph at x=2 is at y=0, so that's wrong. So the correct slope is \( \frac{1}{2} \) and y-intercept -1, so the equation is \( y = \frac{1}{2}x - 1 \). Wait, but let's confirm again. The y-intercept is -1 (so \( b = -1 \)), and slope: from (0, -1) to (2, 0): rise 1, run 2, so slope \( \frac{1}{2} \). So the equation is \( y = \frac{1}{2}x - 1 \).

Answer:

\( y = \frac{1}{2}x - 1 \) (the option: \( y = \frac{1}{2}x - 1 \))