In y = mx + c, m is the gradient — how steeply the line rises or falls — and c is the y-intercept, where the line crosses the y-axis. Once you can read these two values at a glance, you can sketch straight-line graphs quickly and identify parallel lines without plotting a single point.
What does each part of y = mx + c tell you?
- m (gradient): the amount y increases for every 1 unit increase in x. A positive m means the line slopes up from left to right; a negative m means it slopes down. The steeper the line, the larger the magnitude of m.
- c (y-intercept): the value of y when x = 0. It is the point (0, c) where the line crosses the y-axis. A positive c means the crossing is above the origin; a negative c means it is below.
Quick example: y = 3x + 2 has gradient 3 and y-intercept 2. It rises steeply and crosses the y-axis at (0, 2).
How do you read m and c from an equation in standard form?
When the equation is already written as y = mx + c, m and c can be read directly — no rearrangement needed.
| Equation | Gradient (m) | Y-intercept (c) |
|---|---|---|
| y = 2x + 5 | 2 | 5 |
| y = −3x + 1 | −3 | 1 |
| y = ½x − 4 | ½ | −4 |
| y = x + 7 | 1 (coefficient of x is 1) | 7 |
| y = −x | −1 | 0 |
| y = 4 | 0 (no x-term) | 4 |
Notice: a horizontal line y = 4 has gradient 0 because it does not rise or fall. A line written as y = x has m = 1 (the coefficient of x is understood to be 1).
What if the equation is not in y = mx + c form?
Rearrange to make y the subject.
Example: rearrange 3y = 6x − 9 into y = mx + c.
- Divide every term by 3: y = 2x − 3.
- Now m = 2 and c = −3.
Example: rearrange 2x + y = 7.
- Subtract 2x from both sides: y = −2x + 7.
- So m = −2 and c = 7.
A common mistake: writing 2x + 3y = 12 and reading m = 2, c = 12 — these are NOT the gradient and intercept. Always rearrange to y = mx + c first.
How do you identify parallel lines from their equations?
Parallel lines have equal gradients. Check only the coefficient of x — parallel lines may have different y-intercepts, but their m values are identical.
Example: which of these lines are parallel to y = 3x − 1?
- y = 3x + 5 (m = 3) — parallel ✓
- y = 3x − 7 (m = 3) — parallel ✓
- y = −3x + 2 (m = −3) — not parallel (same number, opposite sign)
- 3y = 9x + 1 (rearranges to y = 3x + ⅓, m = 3) — parallel ✓
How do you sketch a straight-line graph from an equation?
- Plot the y-intercept (0, c).
- Use the gradient to find a second point. If m = 2, go 1 right and 2 up. If m = ½, go 2 right and 1 up.
- Draw a straight line through both points, extending in both directions.
Example: sketch y = 2x − 1.
- Y-intercept: (0, −1). Plot this point.
- Gradient 2 means for every 1 right, go 2 up: second point (1, 1).
- Draw the line through (0, −1) and (1, 1).
This method is faster than building a table of values when the equation is in standard form.
Frequently asked questions
What does a gradient of 0 mean?
A gradient of 0 means the line is horizontal — it has no steepness at all. Every point on the line has the same y-value, so the equation is simply y = c (a constant). For example, y = 5 is a horizontal line crossing the y-axis at (0, 5).
How is the gradient related to the ratio "rise over run"?
Gradient = rise ÷ run = (change in y) ÷ (change in x). If m = 3, the line rises 3 units for every 1 unit across. If m = −2, it falls 2 units for every 1 unit across. This is why steeper lines have larger gradients and shallower lines have smaller ones.
Can m or c be a fraction?
Yes. Equations such as y = ¾x + ½ are perfectly valid. To sketch a line with a fractional gradient like ¾, move 4 right and 3 up (or 4 left and 3 down) to avoid dealing with fractions of a unit on the grid.
What is the equation of a vertical line?
A vertical line cannot be written as y = mx + c because its gradient is undefined (the change in x is zero, so rise ÷ run involves dividing by zero). A vertical line is written as x = k, where k is the x-value of every point on it. For example, x = 3 passes through (3, 0), (3, 1), (3, 2), and so on.
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