Question

which equation matches the graph shown below? graph of a line on a coordinate plane options: $y = \frac{4}{3}x - 1$, $y = \frac{4}{3}x - 2$, $y = \frac{3}{4}x - 1$, $y = \frac{3}{4}x - 2$

Explanation:

Step1: Identify the y-intercept

The graph crosses the y-axis at (0, -1) (since it's near y = -1 when x=0). So the equation should have a y-intercept of -1. This eliminates options with -2 (second and fourth).

Step2: Calculate the slope

Take two points on the line, e.g., (0, -1) and (3, 3) (or (10, \(\frac{27}{3}\)? Wait, better to use (0, -1) and (3, 3)? Wait, looking at the grid, from (0, -1) to (3, 3)? Wait, no, let's use (0, -1) and (3, 3)? Wait, actually, let's take (0, -1) and (3, 3)? Wait, no, the line passes through (0, -1) and (3, 3)? Wait, no, let's check the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's take (0, -1) and (3, 3)? Wait, no, looking at the graph, when x=3, y=3? Wait, no, the grid: each square is 1 unit? Wait, the x-axis has 0, 10, 20, 30, so each grid line is 5 units? Wait, no, the graph: the line passes through (0, -1) and (3, 3)? Wait, maybe better to use (0, -1) and (3, 3) is wrong. Wait, let's take (0, -1) and (3, 3) – no, let's look at the options. The slope in the first option is \( \frac{4}{3} \), third is \( \frac{3}{4} \). Let's take two points: (0, -1) and (3, 3) – slope is \( \frac{3 - (-1)}{3 - 0} = \frac{4}{3} \). Wait, but when x=3, y=3? Wait, the graph: when x=3, y=3? Wait, the grid: the x-axis is from -10 to 30, y-axis from -10 to 20. The line passes through (0, -1) and (3, 3)? Wait, no, maybe (0, -1) and (3, 3) is correct. So slope is \( \frac{4}{3} \), y-intercept -1. So the equation is \( y = \frac{4}{3}x - 1 \).

Step3: Match with options

The first option is \( y = \frac{4}{3}x - 1 \), which has slope \( \frac{4}{3} \) and y-intercept -1, matching our findings.

Answer:

A. \( y = \frac{4}{3}x - 1 \)