Doug C. answered • 05/14/25
Math Tutor with Reputation to make difficult concepts understandable
Ordered pairs in the 1st quadrant have x and y coordinates that are both positive. For the given equation y = 3x + 1, x is the independent variable and y is the dependent variable. We say y depends on x. To make sure the ordered pairs satisfying this equation are in the 1st quadrant start by restricting the x values to positive values, then determine if the corresponding y-values are also positive.
Note that when x = 0, y = 3(0) + 1 = 1. That means 1 is the y-coordinate of the y-intercept (0, 1). Since the term containing the variable x has a numerical coefficient of 3 (the slope of the line), the line will slope uphill as the graph goes from left to right. Think of the slope as 3/1 (change in y over change in x). Every time x goes up by 1, the y value increases by 3. This means all y -values where x > 0 must be greater than 1, and positive.
To actually graph the line, place an open circle at (0,1), the y-intercept, then move 1 to the right in the x direction, followed by up 3. That takes you to (1, 4). Try another move 1 right and up 3. Takes you to (2, 7). And so on. Draw a line from the y-intercept through as many other plotted points as you want (only one additional is really required). Make sure your graphed line (ray, actually), goes through the 1st quadrant with an arrow at the end.
You are really graphing y = 3x + 1, x > 0. Visit this graph on Desmos by selecting the link, right-clicking, and choosing "Go to..." from the drop-down menu:
desmos.com/calculator/ytjneosjr5
The graph should start giving you some ideas on stuff that can be done on Desmos.
