TheIBAxolotl
Gradient: the steepness number
The gradient \(m\) of a line measures how much \(y\) changes for each 1-unit step in \(x\). Between two points,
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}.\]
Positive \(m\) goes uphill left to right, negative \(m\) downhill, \(m = 0\) is horizontal, and a vertical line has undefined gradient (its equation is \(x = c\)).
Three ways to write a line
- Gradient-intercept: \(y = mx + c\), where \(c\) is the \(y\)-intercept. Best for reading features.
- Point-gradient: \(y - y_1 = m(x - x_1)\). Best for building a line from a point and a gradient.
- General: \(ax + by + d = 0\). Often requested with integer coefficients.
They are all the same line, rearranged. The \(x\)-intercept is found by setting \(y = 0\); the \(y\)-intercept by setting \(x = 0\).
Parallel and perpendicular
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is \(-1\), so
\[m_2 = -\frac{1}{m_1}.\]
Intuition: rotate a line by \(90^\circ\) and the "rise" and "run" swap places, and one of them changes sign. So a gradient of \(\frac{2}{3}\) becomes \(-\frac{3}{2}\).
Worked example
Points \(A(1, 3)\) and \(B(5, 11)\).
- Gradient: \(m = \frac{11 - 3}{5 - 1} = 2\).
- Line \(AB\): \(y - 3 = 2(x - 1)\), so \(y = 2x + 1\).
- Line through \(B\) perpendicular to \(AB\): gradient \(-\frac12\), so \(y - 11 = -\frac{1}{2}(x - 5)\). Multiply by 2: \(2y - 22 = -x + 5\), giving \(x + 2y - 27 = 0\).
Check: \(B\) satisfies it, since \(5 + 22 - 27 = 0\).
Intersections
Two non-parallel lines meet at exactly one point. Solve the equations simultaneously, by substitution, elimination or with the GDC. For \(y = 2x + 1\) and \(x + 2y - 27 = 0\): substitute to get \(x + 4x + 2 - 27 = 0\), so \(x = 5\), \(y = 11\), which is \(B\), as expected.
In context
In a model like "cost \(= 40 + 12h\)" the gradient 12 is the rate (USD 12 per hour) and the intercept 40 is the fixed charge. Exam questions often ask you to interpret these in words.
Common mistakes
- Subtracting coordinates in different orders on top and bottom of the gradient fraction.
- Using \(-m\) instead of \(-\frac{1}{m}\) for a perpendicular gradient.
- Leaving fractions in the general form when integer coefficients are requested.
- Mixing up the \(x\)- and \(y\)-intercepts.
Exam tips
- Substitute a known point into your final equation as a quick check.
- If the general form is required, clear fractions and move everything to one side; any non-zero multiple is equivalent.
- In context questions, give units when interpreting the gradient.