Understanding the Equation y = mx + c
A linear equation in two variables x and y can be written as y = mx + c, where m and c are constants. Here, m is called the slope (or gradient) of the line, which indicates how steep the line is and the direction it goes. The value c is the y-intercept, which is the point where the line crosses the y-axis (where x = 0).
The slope m tells us how much y changes for a unit increase in x. For example, if m = 2, then for every increase of 1 in x, y increases by 2. If m is negative, the line slopes downwards from left to right.
- m = slope (gradient) of the line
- c = y-intercept (value of y when x = 0)
Plotting the Y-Intercept
The first step in drawing the graph is to plot the y-intercept point on the coordinate axes. Since the y-intercept occurs where x = 0, substitute x = 0 into the equation to find y = c. This gives the point (0, c).
For example, if the equation is y = -2x + 5, then the y-intercept is 5, so plot the point (0, 5) on the y-axis.
- Locate (0, c) on the y-axis
- Mark this point clearly as the starting point of the line
Using the Slope to Find Another Point
The slope m is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It can be written as m = rise/run.
From the y-intercept point, use the slope to find a second point. For example, if m = 3, this means rise = 3 and run = 1. Starting at (0, c), move 1 unit to the right (increase x by 1) and 3 units up (increase y by 3) to find the next point (1, c + 3).
If the slope is negative, move down instead of up. For example, if m = -2, move 1 unit right and 2 units down.
- Slope m = rise/run
- From (0, c), move horizontally by run and vertically by rise
- Plot the second point accordingly
Drawing the Straight Line
Once two points are plotted, use a ruler to draw a straight line passing through both points. Extend the line across the graph in both directions, adding arrowheads to indicate it continues indefinitely.
Label the axes and mark the scale clearly to ensure accuracy. The line represents all solutions (x, y) that satisfy the equation y = mx + c.
- Use a ruler to join the two points
- Extend the line with arrowheads on both ends
- Label axes and mark scale clearly
Worked Example
Draw the graph of y = -2x + 5. Step 1: Identify slope and intercept: m = -2, c = 5. Step 2: Plot y-intercept (0, 5). Step 3: Use slope -2 = -2/1: from (0, 5), move 1 unit right and 2 units down to (1, 3). Step 4: Plot point (1, 3). Step 5: Draw a straight line through (0, 5) and (1, 3) and extend it both ways with arrowheads.
- m = -2, c = 5
- Points: (0, 5) and (1, 3)
- Draw line through points
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify slope and intercept | From y = mx + c, find m and c | y = 3x + 1 → m=3, c=1 | Marks for correct identification |
| Plot y-intercept | Plot point (0, c) on y-axis | (0, 1) for y=3x+1 | Marks for accurate plotting |
| Use slope to find second point | From (0, c), move run horizontally and rise vertically | m=3 → move 1 right, 3 up to (1,4) | Marks for correct use of slope |
| Draw line | Join points with ruler and extend | Line through (0,1) and (1,4) | Marks for neat, straight line with arrowheads |
Active recall
Quick check
1. What is the y-intercept of the line y = 4x - 3?
The y-intercept is -3, so the point (0, -3) on the y-axis.
2. If the slope m = -1/2, how do you find the next point from the y-intercept?
From the y-intercept, move 2 units right (run = 2) and 1 unit down (rise = -1) to find the next point.
3. How do you know if the line slopes upwards or downwards?
If the slope m is positive, the line slopes upwards; if m is negative, it slopes downwards.