Question

what is the equation of this line?
○ $y = -\frac{1}{2}x - 2$
○ $y = -2x - 2$
○ $y = \frac{1}{2}x - 2$
○ $y = 2x - 2$
graph of a line on a coordinate plane

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -2)\) (the y - intercept) and \((4, 0)\) (we can also use other points, but these are easy to identify).

Step2: Calculate the slope \(m\)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0, - 2)\) and \((x_2,y_2)=(4,0)\). Then \(m=\frac{0 - (-2)}{4 - 0}=\frac{2}{4}=\frac{1}{2}\)? Wait, no, wait. Wait, looking at the line, when \(x\) increases, \(y\) decreases? Wait, maybe I picked the wrong points. Let's re - examine. Wait, the line passes through \((0,-2)\) and \(( - 4,0)\)? Wait, no, let's check the direction. Wait, the line is going from the second quadrant to the fourth quadrant, so it has a negative slope. Let's take two points: when \(x = 0\), \(y=-2\) (y - intercept). When \(y = 0\), let's find \(x\). Wait, maybe the line passes through \((0,-2)\) and \((4,0)\)? No, that would be positive slope. Wait, maybe I made a mistake. Wait, the options have negative slopes. Let's recalculate. Let's take two points: \((0,-2)\) and \(( - 4,0)\). Then \(m=\frac{0-(-2)}{-4 - 0}=\frac{2}{-4}=-\frac{1}{2}\).

Step3: Use slope - intercept form \(y=mx + b\)

The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We know that the y - intercept \(b=-2\) (from the point \((0,-2)\)) and the slope \(m =-\frac{1}{2}\). So the equation of the line is \(y=-\frac{1}{2}x-2\).

Answer:

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