Graphs of Equations and Coordinate Geometry

Updated 13 October 2026

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?

Coordinates give every point in the plane a unique address, so a shape can be described and measured without being drawn. The plane is two number lines at right angles, crossing at the origin, (0, 0). A pair (x, y) means go across x, then up y, with negative values going left or down. See the coordinate plane. The point (3, 4) is 3 across and 4 up, and Pythagoras gives its distance from the origin: 3 × 3 + 4 × 4 = 25, which is 5 squared, so the distance is 5. The across and up steps are at right angles, so they form the shorter sides of a right triangle. Order matters: (3, 4) and (4, 3) are different points, and x always comes first. A midpoint averages each coordinate: from (2, 6) to (8, 10), the midpoint is (5, 8), because (2 + 8) ÷ 2 = 5 and (6 + 10) ÷ 2 = 8.

(3, 5) is three across and five up; (5, 3) is a different point entirely.

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

Gradient measures steepness: it is how much y changes for each step of 1 in x. Between two points, divide the change in y by the change in x. From (2, 1) to (6, 9), the gradient is (9 − 1) ÷ (6 − 2) = 2, so the line climbs 2 for every 1 across. See gradient from two points. A negative gradient means y falls as x rises, so the line runs downhill from left to right. From (1, 7) to (4, 1), the change in y is 1 − 7 = −6 over a change in x of 3, so the gradient is −6 ÷ 3 = −2. The sign comes from the subtraction, so take both differences in the same order. A horizontal line has gradient 0, and a vertical line has none, since its change in x is zero. Swapping the order of both points gives the same result: (1 − 9) ÷ (2 − 6) = 2.

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.

What does y = mx + c mean?

In y = mx + c, m is the gradient and c is the y-intercept, where the line crosses the y-axis. For y = 3x − 2, the line crosses the y-axis at (0, −2) and climbs 3 for every 1 across. Putting x = 0 leaves y = c, and each step of 1 in x adds m to y. See y = mx + c. To find the equation from two points, work out m, then substitute a point to find c. Through (1, 5) and (3, 9), m is (9 − 5) ÷ (3 − 1) = 2, and (1, 5) gives 5 = 2 + c, so c = 3 and the line is y = 2x + 3. Another form must be rearranged first: 2y = 4x + 6 has gradient 2, not 4, once both sides are divided by 2. Test the equation with the second point: 2 × 3 + 3 = 9.

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?

1. Find the gradient from the two points.
2. Put that gradient and either point into $y-y_1=m\left(x-x_1\right)$y − y₁ = m(x − x₁).
3. 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?

Two lines are parallel when their gradients are equal, and perpendicular when their gradients multiply to −1. So y = 2x + 1 and y = 2x − 5 are parallel: both climb 2 for every 1 across. A line perpendicular to them has gradient $-\frac{1}{2}$−1/2, because $2\times -\frac{1}{2}=-1$2 × −1/2 = −1. See gradients of parallel and perpendicular lines. Turning a line through a right angle swaps its rise and run and changes one sign: 1 across and 2 up becomes 2 across and 1 down. So flip the fraction and change its sign, and $\frac{3}{4}$3/4 becomes $-\frac{4}{3}$−4/3. A horizontal and a vertical line are perpendicular, but the product rule cannot show it, since a vertical line has no gradient. The line through (4, 1) perpendicular to y = 2x + 1 is $y=-\frac{x}{2}+3$y = −x/2 + 3, since $-\frac{1}{2}\times 4+3=1$−1/2 × 4 + 3 = 1. Multiply a pair of gradients to confirm: $\frac{3}{4}\times -\frac{4}{3}=-1$3/4 × −4/3 = −1.

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?

Most graphs belong to a few standard families, and each can be recognized from its equation before you plot a point. A straight line has no power of x above 1. A quadratic, with an $x^2$x² term, is a parabola: symmetric, with one turning point, opening upward when that coefficient is positive. See quadratic graphs. The curve $y=x^2-4$y = x² − 4 turns at (0, −4) and crosses the x-axis at 2 and −2, since 2 × 2 − 4 = 0. A cubic can bend twice and runs from one corner of the plane to the opposite one; see quadratic and cubic graphs. The highest power sets the shape, because for large x it outgrows the other terms. A negative leading coefficient flips the picture: $y=-x^2$y = −x² opens downward. Plotting a few points confirms a sketch. On the same parabola, x = 3 gives 3 × 3 − 4 = 5, and x = −3 gives the same value, as the symmetry says.

Reciprocal graphs are the first curve with a hole in the middle of it: $y=\frac{1}{x}$y = 1/x has no value at x = 0 and hugs both axes without ever touching them. That behavior 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?

To find where two graphs cross, set their equations equal and solve: a crossing point satisfies both equations at once. For y = 2x + 1 and y = 7 − x, setting 2x + 1 = 7 − x gives 3x = 6, so x = 2, and either equation gives y = 5. See simultaneous equations for the algebra and intersections of graphs for reading a sketch. That point lies on both graphs. The number of solutions is the number of crossings. When one graph is a curve, the algebra gives a quadratic, with two crossings, one or none: $y=x^2$y = x² and y = x + 2 meet where $x^2-x-2=0$x² − x − 2 = 0, at x = 2 and x = −1, the points (2, 4) and (−1, 1). A crossing read from a graph is only an estimate, so confirm it in both equations: 2 × 2 + 1 = 5 and 7 − 2 = 5.

What is the equation of a circle?

A circle of radius r centered at (a, b) has the equation ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$(x − a)² + (y − b)² = r². It is Pythagoras: for any point (x, y) on the circle, the horizontal and vertical distances to the center are the shorter sides of a right triangle, and the radius is the hypotenuse. See the equation of a circle. So the circle centered at (2, −1) with radius 5 is ${\left(x-2\right)}^2+{\left(y+1\right)}^2=25$(x − 2)² + (y + 1)² = 25. Each bracket's sign is the opposite of the center's coordinate, a common slip. The steps change when the equation arrives expanded, as in $x^2+y^2-4x+2y-20=0$x² + y² − 4x + 2y − 20 = 0: complete the square on the x terms and on the y terms to get back to the bracket form. Test a point you expect on the circle: (5, 3) is 3 across and 4 up from the center, and 3 × 3 + 4 × 4 = 25.

Another expanded circle is $x^2+y^2-6x+4y-3=0$x² + y² − 6x + 4y − 3 = 0, and completing the square finds its center and radius.

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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$x²/a² + y²/b² = 1, where a and b are the distances from the center 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: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$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 toward a pair of straight-line asymptotes. That is the same escape-to-infinity behavior 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 traveled — 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 toward 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 ${\left(x-2\right)}^2+{\left(y+1\right)}^2=25$(x − 2)² + (y + 1)² = 25 the radius is 5, not 25.
• Reading the signs of the center off the brackets directly. ${\left(x-2\right)}^2$(x − 2)² means the center is at x = +2.

From here the two natural neighbors 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. Related guides: Graph Theory, Networks and Voronoi Diagrams and Scatter Plots, Correlation and Regression.