Final answer:
To find additional points on a line, we use the given point and slope to determine the rise over run. Zero slope indicates a horizontal line, while an undefined slope indicates a vertical line. An example is given for each case to illustrate how to find three additional points on the line.
Step-by-step explanation:
To find additional points on a line given a starting point and the slope, we can use the concept of slope as rise over run. For each of the cases provided, we'll apply this principle to find three more points that lie on the same line.
Point (2,1) with slope m=0: Since the slope is zero, the line is horizontal. Additional points can be (3,1), (4,1), and (5,1).
Point (3,-2) with slope m=0: Similar to the previous case, the line is horizontal. Additional points would be (4,-2), (5,-2), and (6,-2).
Point (1,5) with slope m undefined: An undefined slope means the line is vertical. More points on this line are (1,6), (1,7), and (1,8).
Point (-4,1) with slope m undefined: As the line is vertical, additional points are (-4,2), (-4,3), and (-4,4).
Point (0,-9) with slope m=-2: This negative slope means that for a move 1 unit right (positive run), the line goes down 2 units (negative rise). More points are (1,-11), (2,-13), and (3,-15).
Point (-5,4) with slope m=4: A rise of 4 for every 1 unit of run. Additional points are (-4,8), (-3,12), and (-2,16).
Point (7,-2) with slope m=1/2: This slope indicates a rise of 1 for every 2 units of run. Additional points could be (9,-1), (11,0), and (13,1).
Point (-1,-6) with slope m=-1/3: A fall of 1 unit for every 3 units of rightward run (negative rise over positive run). More points are (2,-7), (5,-8), and (8,-9).
Through these calculations, you can see that specifying the y-intercept and slope of a line allows you to identify additional points through which the line passes, defining the shape of the line.