Gradient and Intercept
The equation y = mx + c describes any straight line. Understanding the roles of the gradient 'm' and the y-intercept 'c' allows you to instantly interpret, sketch, and transform linear graphs.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The value 'c' is the y-intercept, which is the point where the line crosses the vertical y-axis. Its coordinate is (0, c). Altering 'c' shifts the entire line vertically up or down.
- The value 'm' is the gradient, which measures the steepness of the line. It represents the change in y for every one-unit increase in x.
- A positive gradient (m > 0) indicates a line that slopes upwards from left to right.
- A negative gradient (m < 0) indicates a line that slopes downwards from left to right.
- The magnitude (absolute value) of m determines the steepness. A line with m = -5 is steeper than a line with m = 2.
- If m = 0, the equation becomes y = c, which is a horizontal line.
Diagram
Formulae
y = mx + c To describe the relationship between the x and y coordinates for any point on a straight line.
m = (y2 - y1) / (x2 - x1) To calculate the gradient of a line that passes through two known points, (x1, y1) and (x2, y2).
Definitions
- Gradient (m)
- A measure of a line's steepness, calculated as the ratio of vertical change ('rise') to horizontal change ('run') between any two points on the line.
- y-intercept (c)
- The point where the line intersects the y-axis. At this point, the x-coordinate is always zero.
Worked example
The line L1 has the equation y = 4x - 5. A new line, L2, is formed by halving the gradient of L1 and increasing its y-intercept by 8. What is the equation of L2?
- 1
First, identify the gradient (m) and y-intercept (c) of the original line L1.
For y = 4x - 5, we have m1 = 4 and c1 = -5 - 2
Next, calculate the new gradient for L2.
The gradient is halved, so m2 = m1 / 2 = 4 / 2 = 2.
- 3
Then, calculate the new y-intercept for L2.
The intercept is increased by 8, so c2 = c1 + 8 = -5 + 8 = 3.
- 4
Finally, construct the equation for L2 using the new gradient and y-intercept.
The equation is y = 2x + 3.
Answer: y = 2x + 3
Common mistakes
- ×Mistaking the y-intercept 'c' for the x-intercept. The value of 'c' tells you where the line crosses the vertical axis only.
- ×Incorrectly comparing the steepness of lines with negative gradients. For example, a line with gradient -3 is steeper than one with gradient -1, because its absolute value is larger.
- ×Thinking that changing the y-intercept 'c' also changes the gradient 'm'. Shifting a line up or down does not affect its steepness.
No-calculator tips
- ✓To quickly sketch a line, first mark the y-intercept (0, c) on the y-axis. Then, use the gradient m as 'rise over run'. For m = 3, move 1 unit right and 3 units up to find a second point.
- ✓To find the x-intercept without rearranging the whole equation, mentally set y = 0 and solve for x. The x-intercept is always at x = -c/m.
- ✓If two equations have the same value for 'm', the lines are parallel. This means they will never intersect.