Why does a negative gradient mean the line goes down?

Because gradient is the change in y divided by the change in x, and moving right means the change in x is positive. If the fraction is negative, the change in y must therefore be negative — so as you go right, you go down. A gradient of minus 3 says that every step right is three steps down, which is the same steepness as plus 3, just tilted the other way.

Why do perpendicular gradients multiply to minus 1?

Rotating a line by a right angle swaps the horizontal and vertical steps and reverses one of their signs. A run of 2 and a rise of 1 becomes a run of minus 1 and a rise of 2, so a gradient of one half becomes minus 2. Multiplying the old gradient by the new one always gives minus 1, because you have flipped the fraction and changed the sign.

How do you tell an ellipse from a hyperbola?

Look at the sign between the two squared terms. An ellipse has a plus — x squared over a squared plus y squared over b squared equals 1 — and is a closed oval. A hyperbola has a minus and is two separate open branches that head off towards a pair of straight-line asymptotes. Both reduce to a circle-like symmetry, but the minus sign is what lets the curve escape to infinity.

What does the area under a speed-time graph mean?

It is the distance travelled. Speed multiplied by time gives distance, and on a graph a height multiplied by a width is an area, so the two are the same calculation. The gradient of the same graph gives acceleration, because that is a change in speed divided by a change in time. One graph therefore answers two different questions depending on whether you measure its slope or its area.

Graphs of Equations and Coordinate Geometry

Q: How do you find the equation of a line through two points?

Find the gradient first by dividing the difference in the y values by the difference in the x values, then substitute either point into y minus y1 equals m times x minus x1. For the points (1, 4) and (3, 10), the gradient is 6 over 2, which is 3, and the equation becomes y minus 4 equals 3 times x minus 1, or y equals 3x plus 1. Using the other point gives the same line, which is a useful check.

Q: What is the equation of a circle?

A circle of radius r centred at the point (a, b) has the equation (x minus a) squared plus (y minus b) squared equals r squared. It is Pythagoras' theorem written as an equation: the left-hand side is the squared distance from the centre to the point (x, y), and a circle is precisely the set of points whose distance from the centre is r.

Coordinate geometry is one idea repeated: a curve is the set of points that satisfy an equation. Once a shape has an equation, questions about the shape become questions about algebra, and algebra can be checked.

That is the whole trade Descartes proposed. Distance becomes Pythagoras, "the line is a tangent" becomes "the discriminant is zero", and a geometric argument that needed a clever construction becomes a calculation anyone can carry out.

What do coordinates actually do?

They give every point in the plane a unique address, so a shape can be described without being drawn. The coordinate plane sets this up: two number lines at right angles, and a pair (x, y) reached by going across then up.

The order matters and always trips people once. (3, 5) is three across and five up; (5, 3) is a different point entirely. Across before up, in the same order as the alphabet puts x before y.

What is gradient, and why does a negative gradient go downhill?

Gradient is how much y changes for each step of 1 in x, so a negative gradient means y falls as x rises — the line goes downhill left to right. Gradient from two points computes it as the difference in the y values divided by the difference in the x values.

From (1, 4) to (3, 10): the rise is 6, the run is 2, so the gradient is 3. From (0, 7) to (2, 1): the rise is −6, the run is 2, so the gradient is −3.

The sign is not a separate rule that has to be remembered. Going right makes the run positive, so the sign of the fraction is entirely decided by whether y went up or down. A gradient of −3 and a gradient of 3 describe equally steep lines; the minus is a direction, not a size. This is also why swapping which point you call "first" changes nothing — both differences flip sign, and the fraction is unmoved.

What does y = mx + c mean?

m is the gradient and c is where the line crosses the y-axis, so the two numbers between them pin down the line completely. y = mx + c is the form to reach for when you want to sketch a line or compare two of them at a glance.

It is not the only form, and the others are not decoration. Writing a line as Ax + By = C handles the one case y = mx + c cannot — a vertical line, whose gradient is undefined — and is the shape simultaneous equations naturally arrive in. Point-slope form is the fastest route when you know a gradient and one point, which is the situation almost every tangent question hands you.

Those three forms are one line wearing different clothes, and coordinate geometry is the lesson that ties them together before the harder work starts.

How do you find the equation of a line through two points?

Find the gradient from the two points.
Put that gradient and either point into y − y₁ = m(x − x₁).
Rearrange into whichever form the question wants.

Through (1, 4) and (3, 10): gradient 3, so y − 4 = 3(x − 1), giving y = 3x + 1. The equation of a line through two points works this through, including the check worth doing every time — substitute the other point and confirm it fits.

When are two lines parallel or perpendicular?

Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1. Gradients of parallel and perpendicular lines is short and does more work than any other lesson on this page, because half of coordinate geometry reduces to one of those two tests.

Turn a line a quarter turn and its rise becomes its run. A step of 2 across and 1 up becomes a step of 1 across the other way and 2 up — so a gradient of ½ becomes −2. You have inverted the fraction and flipped the sign, and doing both to a number and multiplying gives −1 every time. The one exception is a horizontal line meeting a vertical one, where the gradient of the vertical line does not exist to be multiplied.

Midpoints and distances

The midpoint is the average of the two points, taken coordinate by coordinate. The midpoint of a segment makes the reason plain: the halfway point across is the average of the x values, and the same holds going up.

Distance is Pythagoras with the right-angled triangle left implicit. The distance between two points builds it: the horizontal gap and the vertical gap are the two short sides, and the straight-line distance is the hypotenuse. From (1, 2) to (4, 6): 3 across, 4 up, so the distance is 5.

What do the standard curves look like?

Four shapes cover most of what you will meet, and each is recognisable from its equation before you plot a single point. Quadratic graphs introduces the parabola — symmetric, with one turning point, opening upwards when the x² coefficient is positive — and quadratic and cubic graphs adds the cubic, which can wiggle twice and always runs from one corner of the plane to the opposite one.

Reciprocal graphs are the first curve with a hole in the middle of it: y = 1/x has no value at x = 0 and hugs both axes without ever touching them. That behaviour is the reader's first meeting with an asymptote, and it comes back in force in rational functions.

The modulus curves are the ones that look strange and are not. The graph of y = |x| is a V, because the modulus leaves positive inputs alone and reflects negative ones upward. Taking the modulus of a whole expression applies that reflection to any curve — whatever was below the axis flips above it — while replacing x with a modulus does something quite different, throwing away the left half of the graph and replacing it with a mirror image of the right. Two similar-looking notations, two unrelated pictures.

How do you find where two graphs cross?

Set the equations equal and solve — a crossing point is a pair (x, y) that satisfies both equations at once, which is exactly what simultaneous equations mean. Simultaneous equations handles the algebra and intersections of graphs handles the reading: how many solutions the algebra produces is how many times the curves cross.

This is why "no real solution" is not a failure of the method. Two graphs that miss each other cannot produce an intersection, and the algebra says so honestly by handing back a negative discriminant. The algebra and the picture never disagree; when they seem to, one of them has been read wrong.

What is the equation of a circle?

A circle of radius r centred at (a, b) is (x − a)² + (y − b)² = r², which is Pythagoras dressed as an equation. The equation of a circle derives it from the distance formula, and once you see that the left-hand side is a squared distance the form stops needing to be memorised.

Real questions rarely hand it over in that shape. Expanded, it arrives as something like x² + y² − 6x + 4y − 3 = 0, and getting the centre and radius back means completing the square on the x terms and the y terms separately — the same technique used to solve quadratics, put to a geometric purpose.

Two follow-up questions come up constantly. The tangent to a circle at a point uses one fact — a tangent is perpendicular to the radius at the point of contact — so the tangent's gradient is the negative reciprocal of the radius' gradient, and no calculus is needed. Where a line meets a circle substitutes the line into the circle and reads the discriminant of the quadratic that falls out: two roots for a chord, one for a tangent, none for a miss.

Ellipses and hyperbolas

An ellipse is a circle that has been stretched by different amounts in each direction, and its equation records both stretches. The equation of an ellipse gives x²/a² + y²/b² = 1, where a and b are the distances from the centre to the curve along each axis; setting a = b recovers the circle exactly.

Change that plus to a minus and the curve breaks in two. The equation of a hyperbola explains why: x²/a² − y²/b² = 1 has no solutions at all for small x, so a whole band down the middle of the plane is empty, and the two branches that remain run away towards a pair of straight-line asymptotes. That is the same escape-to-infinity behaviour reciprocal graphs show, in a more symmetric setting.

Graphs that describe a journey

On a distance-time graph the gradient is speed, because a change in distance divided by a change in time is exactly what speed means. Distance-time graphs reads them the other way as well: a flat section is a stop, and a steeper line is a faster stretch.

Speed-time graphs change what the features mean. Here the gradient is acceleration and the area underneath is the distance travelled — a rectangle of speed × time is a distance, and the same holds for any shape underneath the curve.

The general version of this is the average rate of change over an interval: the gradient of the straight line joining two points on any curve, which says how fast the quantity changed on average between them. Shrink that interval towards nothing and you have the central idea of differential calculus, which is where this thread continues.

The mistakes worth naming

Reading the intercept as the gradient. In y = 4 + 2x the gradient is 2, not 4. Rearranging into y = mx + c first removes the ambiguity.
Subtracting the coordinates in opposite orders. Both differences must run the same way round, or the gradient comes out with the wrong sign.
Taking the radius as the number on the right. In (x − 2)² + (y + 1)² = 25 the radius is 5, not 25.
Reading the signs of the centre off the brackets directly. (x − 2)² means the centre is at x = +2.

Learn this properly in the app

Math Challenge teaches all of this as illustrated lessons with try-it problems, inside a catalog of 800+ lessons from counting to calculus. From here the two natural neighbours are quadratics and polynomials, which is the algebra behind every parabola on this page, and functions and rational functions, which takes the transformation ideas further.

Your turn

Three to try — tap what you get.

The gradient of y = 3x − 2

Where does y = 2x + 1 cross the y-axis?

The midpoint of (2, 4) and (6, 8)

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