Question

graph the line. y = -\frac{1}{3}x + 2

Explanation:

Step1: Identify the y - intercept

The equation of the line is in the slope - intercept form \(y = mx + b\), where \(b\) is the y - intercept. For the equation \(y=-\frac{1}{3}x + 2\), when \(x = 0\), \(y=2\). So the line passes through the point \((0,2)\).

Step2: Identify the slope

The slope \(m=-\frac{1}{3}\). The slope is defined as \(\frac{\text{rise}}{\text{run}}\). A slope of \(-\frac{1}{3}\) means that for every 3 units we move to the right (run = 3) along the x - axis, we move down 1 unit (rise=- 1) along the y - axis. Starting from the y - intercept \((0,2)\), if we move 3 units to the right (to \(x = 3\)) and 1 unit down (to \(y=2 - 1=1\)), we get the point \((3,1)\). We can also move in the opposite direction: for every 3 units we move to the left (run=-3), we move up 1 unit (rise = 1). Starting from \((0,2)\), moving 3 units to the left (to \(x=- 3\)) and 1 unit up (to \(y = 2+1 = 3\)) gives the point \((-3,3)\).

Step3: Plot the points and draw the line

Plot the points \((0,2)\), \((3,1)\) (or \((-3,3)\)) on the coordinate plane. Then, use a straight - edge to draw a line passing through these points. The line should extend in both directions to represent the infinite nature of a line.

(Note: Since this is a graphing problem, the final answer is the graph of the line \(y =-\frac{1}{3}x + 2\) passing through the points \((0,2)\) and other points determined by the slope, such as \((3,1)\) or \((-3,3)\) and extending infinitely in both directions.)

Answer:

The line is graphed by plotting the y - intercept \((0,2)\) and using the slope \(-\frac{1}{3}\) to find additional points (e.g., \((3,1)\) or \((-3,3)\)) and then drawing a straight line through these points.