Find the gradient of a straight line as rise over run, interpret the sign of the gradient and what steepness means, read the y-intercept from a graph, and calculate the gradient between two points

What this dot point is asking

The gradient of a straight line measures how steeply it slopes, and it is the single most important number a line carries. NESA wants you to find it two ways, read its meaning, and pair it with the line's starting value. Four skills are bundled here, and a question can test any of them: find a gradient from a graph by counting the rise over the run; find the gradient between two given points using the formula; read the sign of a gradient (positive, negative, zero or undefined) and say what its size tells you about steepness; and read the $y$-intercept straight off a graph. In a real context the gradient is always a rate (dollars per hour, dollars per kilometre, metres climbed per metre across) and the $y$-intercept is always the starting value (a fixed fee, a flagfall, a height at the start). So interpreting them in words is part of the dot point too.

The deeper idea worth carrying through all of it is that the gradient is a rate of change: it says how much $y$ moves for each one-unit step in $x$. That is why a steeper line means a faster change, a negative gradient means $y$ is falling as $x$ grows, and a flat line (zero gradient) means $y$ never changes. The marks come from a correctly drawn rise/run triangle, subtracting the coordinates in a consistent order, dividing rise by run the right way up, and stating the sign and units. Marks are lost in predictable ways. The usual slips are dividing run by rise, subtracting the two points in opposite orders top and bottom, reading a falling line as positive, and confusing the $y$-intercept with the $x$-intercept.

The answer

Gradient as rise over run

The gradient of a line, usually called $m$, is the vertical rise divided by the horizontal run between any two points on it:

To find it from a graph, pick two points where the line passes cleanly through grid corners, then build a right-angled triangle: go across from the first point to under the second (that horizontal distance is the run), then up or down to the second point (that vertical distance is the rise). Divide the rise by the run. Because a straight line has the same steepness everywhere, it does not matter which two points you choose or how big you make the triangle: the ratio is always the same, which is exactly what makes the gradient a single number for the whole line.

The line below passes through $(1, 1)$ and $(5, 4)$. Going across from $(1, 1)$ to $(5, 1)$ is a run of $4$, and going up from $(5, 1)$ to $(5, 4)$ is a rise of $3$, so the gradient is $\tfrac{3}{4}$. A bigger triangle, say from $(1, 1)$ all the way across, would give the same ratio.

So $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{3}{4} = 0.75$. The order matters less than the consistency: as long as you go across and up in the same direction, the signs look after themselves.

The sign of the gradient: positive, negative, zero and undefined

The sign of the gradient tells you which way the line tilts, before you even look at how steep it is. Read every line left to right (the direction of increasing $x$) and ask what $y$ is doing.

• A positive gradient ($m > 0$) means the line goes up to the right: as $x$ increases, $y$ increases. The rise and the run have the same sign.
• A negative gradient ($m < 0$) means the line goes down to the right: as $x$ increases, $y$ decreases. A positive run pairs with a negative rise.
• A zero gradient ($m = 0$) means a horizontal line: there is no rise, so $\tfrac{0}{\text{run}} = 0$ and $y$ never changes. These are the lines $y = k$.
• An undefined gradient means a vertical line: there is no run, so $\tfrac{\text{rise}}{0}$ asks you to divide by zero, which is not allowed. These are the lines $x = k$. We say the gradient is undefined, not zero or infinite.

The three panels below show the first three cases on identical axes, so you can compare the tilt at a glance.

A vertical line is the fourth case and does not fit a panel like these, because it would be a single upright stroke with no run; remember it separately as the undefined gradient, the line $x = k$.

What steepness means: the size of the gradient

While the sign tells you the direction, the size of the gradient (its value ignoring any minus sign) tells you the steepness. A larger size is a steeper line, because $y$ changes more for each step across. A gradient of $3$ climbs three times as fast as a gradient of $1$; a gradient of $\tfrac{1}{2}$ is gentle, rising only half a unit per unit across. To compare steepness, compare the absolute values: a line of gradient $-3$ is steeper than a line of gradient $2$, because $|-3| = 3$ is bigger than $|2| = 2$, even though one falls and one rises. This is why a wheelchair ramp built to the Australian standard of at most $1$ in $14$ (a gradient of about $0.07$) looks almost flat, while a steep driveway at $1$ in $5$ (a gradient of $0.2$) is noticeably harder to climb: the bigger the number, the steeper the slope.

Reading the y-intercept from a graph

The $y$-intercept is the value of $y$ where the line crosses the vertical axis, the point $(0, b)$. You read it straight off a graph by finding where the line cuts the $y$-axis: no calculation needed. The letter $b$ is the usual name for it. In a real context the $y$-intercept is the starting value of the relationship, what $y$ equals when $x$ is zero: a fixed monthly fee before you use any data, a taxi flagfall before the car moves, a tank's level before you start draining it.

Reading it correctly means looking at the vertical axis. A frequent slip is to read the $x$-intercept (where the line crosses the horizontal axis) by mistake; the $y$-intercept is always the crossing on the up-and-down axis. The diagram below shows a phone-plan cost line: it cuts the vertical axis at $(0, 20)$, so the $y$-intercept is $20 dollars, the fixed fee. The same line lets you read its gradient with a rise/run triangle: from$x = 10$to$x = 30$the run is$20$GB and the cost rises from$25 dollars to $35 dollars, a rise of$10 dollars, giving $m = \tfrac{10}{20} = 0.5$, that is, $0.50 per GB.

Finding the gradient between two points

When you are given two points rather than a graph, you do not need to plot anything: the gradient formula computes the rise over run straight from the coordinates. For points $(x_1, y_1)$ and $(x_2, y_2)$,

The top is the rise (the difference in the $y$-values) and the bottom is the run (the difference in the $x$-values). The one rule that prevents almost every error is to subtract in the same order top and bottom: if you start with the second point's $y$ on top, start with the second point's $x$ on the bottom. Take $(2, 56)$ and $(5, 140)$ for a worker's pay against hours:

The gradient is $28$, which in context is the pay rate of $28 dollars per hour. It would not matter if you subtracted the other way,$\tfrac{56 - 140}{2 - 5} = \tfrac{-84}{-3} = 28$, because both signs flip and cancel; what breaks it is mixing the orders,$\tfrac{140 - 56}{2 - 5} = \tfrac{84}{-3} = -28$, which gives the wrong sign.

How exam questions ask about gradient and intercept

The wording tells you which skill to reach for:

• "Find the gradient of the line" with a graph means draw a rise/run triangle between two grid points and divide rise by run.
• "Find the gradient of the line through $(\ldots)$ and $(\ldots)$" means use the formula $m = \tfrac{y_2 - y_1}{x_2 - x_1}$, keeping the subtraction order consistent.
• "State whether the gradient is positive or negative" (or "is the line increasing or decreasing?") is testing the sign: up to the right is positive and increasing, down to the right is negative and decreasing.
• "Which line is steeper?" is testing the size: compare the gradients ignoring their signs.
• "Write down the $y$-intercept" or "where does the line cut the $y$-axis?" means read the value off the vertical axis at $(0, b)$.
• "What does the gradient represent?" or "interpret the $y$-intercept in this context" wants words: the gradient is the rate (with units like dollars per hour) and the intercept is the starting value (a fixed fee or initial amount).

Why the order of subtraction does not change the gradient

It is worth seeing why subtracting the two points in either order gives the same gradient. This is the reasoning that stops the most common formula error from worrying you. The gradient is a ratio of two differences, and if you swap which point comes first, both the top and the bottom change sign:

In the second version the top is the negative of the first top and the bottom is the negative of the first bottom, so the two minus signs cancel and the value is identical. What you must not do is take $y_2 - y_1$ on top but $x_1 - x_2$ on the bottom: then only one sign flips, and the gradient comes out with the wrong sign, turning an increasing line into a decreasing one. The safe habit is to write the second point's coordinates first in both the top and the bottom, every time. There is one edge case the formula handles automatically and one it cannot. A horizontal line has $y_2 = y_1$, so the top is $0$ and $m = 0$ as expected. A vertical line has $x_2 = x_1$, so the bottom is $0$, and dividing by zero is undefined, which is exactly why a vertical line has no gradient.