Gradient \(= \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in } y}{\text{change in } x}\). Uphill from left to right is positive, downhill is negative, a horizontal line has gradient 0, and a vertical line has no gradient (it is undefined).
Fig. 1: from A(0, 1) to B(4, 7) the line rises 6 for a run of 4, so its gradient is 6 / 4 = 1.5
Section 01
Rise over run.
Drag the two points in the lab and watch the gradient triangle. Make the line steeper, then flatter, then slope it downhill. Try swapping the points round: does the gradient change? Finally drag one point straight above the other and see what happens.
Draw a triangle under the line between two points where the line crosses grid corners exactly.
Run = how far across (left to right). Rise = how far up (negative if it goes down).
Gradient = rise ÷ run. A bigger number means a steeper line.
Direction: uphill (left to right) is positive, downhill is negative, horizontal is 0, vertical is undefined.
Worked example Uphill line from a grid
A straight line passes through the grid points \((0, 1)\) and \((4, 7)\). Find its gradient.
Run: from \(x = 0\) to \(x = 4\) is 4 units across.
Rise: from \(y = 1\) to \(y = 7\) is 6 units up.
Gradient \(= \dfrac{6}{4} = \dfrac{3}{2}\).
Answer: \(\dfrac{3}{2}\) or \(1.5\) (Fig. 1)
Worked example Downhill line from a grid
A straight line passes through \((0, 4)\) and \((2, 0)\). Find its gradient.
Going left to right, the run is 2 units.
The line goes DOWN 4 units, so the rise is \(-4\).
Gradient \(= \dfrac{-4}{2} = -2\).
Answer: \(-2\)
Examiner trap
Counting squares when the axes have different scales. If one square across is 2 units and one square up is 10 units, read the actual values off the axes, not the number of squares.
Section 02
Two points, one formula.
(Extended) When there is no grid, use the coordinates. The change in \(y\) is the rise and the change in \(x\) is the run, so for points \((x_1, y_1)\) and \((x_2, y_2)\): \[m = \dfrac{y_2 - y_1}{x_2 - x_1}\]
Worked example Negative coordinates (Extended)
Find the gradient of the line joining \((-2, 5)\) and \((4, -7)\).
Change in \(y\): \(-7 - 5 = -12\).
Change in \(x\): \(4 - (-2) = 6\).
\(m = \dfrac{-12}{6} = -2\).
Answer: \(-2\)
Worked example A fractional gradient (Extended)
Find the gradient of the line through \((3, -1)\) and \((7, 1)\).
Change in \(y\): \(1 - (-1) = 2\).
Change in \(x\): \(7 - 3 = 4\).
\(m = \dfrac{2}{4} = \dfrac{1}{2}\).
Answer: \(\dfrac{1}{2}\)
Your turn
(Extended) Find the gradient of the line through \((-3, -4)\) and \((1, 8)\).
Taking the \(y\) values in one order and the \(x\) values in the other order. If you start with \(y_2\) on top, you must start with \(x_2\) underneath, or the sign flips. And it is \(y\) over \(x\), never \(x\) over \(y\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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On a grid, a line goes up 6 units for every 3 units across. What is its gradient?
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A straight line slopes downwards from left to right. Its gradient is
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What is the gradient of the line \(y = 4\)?
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(Extended) What is the gradient of the line through \((1, 2)\) and \((3, 8)\)?
Section 03
What the gradient tells you.
A gradient is a rate: how much \(y\) changes for each 1 unit of \(x\). On a distance-time graph it is a speed; on a cost graph it is a price per unit. In the Extended course you also use it backwards, to find a missing coordinate.
Worked example Gradient as a rate
The cost of a phone call is shown by a straight line from \((0, 10)\) to \((100, 25)\), where \(x\) is the time in minutes and \(y\) is the cost in dollars. Find the gradient and say what it means.
Set it equal to 3: \(\dfrac{11 - k}{3} = 3\), so \(11 - k = 9\).
So \(k = 2\). Check: \(\dfrac{11 - 2}{3} = 3\).
Answer: \(k = 2\)
Examiner trap
A horizontal line has gradient 0 (no rise at all), but a VERTICAL line does not have gradient 0: its run is 0, and you cannot divide by 0, so its gradient is undefined.
Stage 2 · Exam
Exam-style questions
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(Extended) Find the gradient of the line joining \(P(-3, 7)\) and \(Q(5, -5)\).
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A straight line on a grid passes through \((0, -3)\) and \((2, 5)\). What is its gradient?
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(Extended) The line through \((2, k)\) and \((6, 1)\) has gradient \(-2\). Find \(k\).
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The depth of water in a tank falls steadily. A straight-line graph joins \((0, 120)\) and \((40, 20)\), where \(x\) is the time in minutes and \(y\) is the depth in cm. At what rate does the depth fall?
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