Question

write the equation of the line in fully simplified slope - intercept form.

Explanation:

Step1: Identify slope and intercept

The line crosses the y - axis at (0, - 2), so the y - intercept \(b=-2\). To find the slope \(m\), use two points. Let's take (0, - 2) and (4, - 1). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So \(m=\frac{-1-(-2)}{4 - 0}=\frac{-1 + 2}{4}=\frac{1}{4}\)? Wait, no, looking at the graph again. Wait, maybe I misread the points. Wait, the line passes through (0, - 2) and (8, 0)? Wait, no, let's check the graph. Wait, the x - axis and y - axis: when x = 0, y=-2 (y - intercept). Then, when y = 0, x = 4? Wait, no, maybe the points are (0, - 2) and (4, - 1)? Wait, no, let's do it correctly. Let's take two clear points. The line goes through (0, - 2) and (4, - 1)? Wait, no, maybe (0, - 2) and (8, 0). Wait, if x = 8, y = 0, and x = 0, y=-2. Then slope \(m=\frac{0-(-2)}{8 - 0}=\frac{2}{8}=\frac{1}{4}\)? No, that doesn't seem right. Wait, maybe the line has a slope of \(\frac{1}{2}\)? Wait, no, let's look again. Wait, the graph: when x increases by 4, y increases by 2? Wait, from (0, - 2) to (4, 0)? Wait, no, the y - intercept is at (0, - 2), and when x = 4, y = 0? Then slope \(m=\frac{0-(-2)}{4 - 0}=\frac{2}{4}=\frac{1}{2}\). Ah, yes, that makes sense. So slope \(m=\frac{1}{2}\), y - intercept \(b = - 2\).

Step2: Write the equation

The slope - intercept form is \(y=mx + b\). Substituting \(m=\frac{1}{2}\) and \(b=-2\), we get \(y=\frac{1}{2}x-2\). Wait, no, wait, maybe I made a mistake. Wait, let's check another point. If x = 2, y=-1. So \(y=\frac{1}{2}x-2\): when x = 2, \(y=\frac{1}{2}(2)-2=1 - 2=-1\), which matches. When x = 4, \(y=\frac{1}{2}(4)-2=2 - 2 = 0\), which also matches. So the slope is \(\frac{1}{2}\) and y - intercept is - 2.

Answer:

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