Graphs in context, gradients, areas and circles

Maths · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A graph of a real situation tells a story, and two measurements carry most of it: the gradient and the area under the graph.

The gradient is a rate of change. On a distance-time graph it is speed; on a velocity-time graph it is acceleration. On a curve the gradient keeps changing, so draw a tangent at the point and find the gradient of that line. The area under a velocity-time graph is the distance travelled.

A circle with its centre at the origin has equation x² + y² = r². A tangent touches the circle at one point and is perpendicular to the radius at that point.

Rules to remember

Distance-time graph: gradient = speed. A horizontal line means not moving. A steeper line means faster.
Velocity-time graph: gradient = acceleration (m/s²). A horizontal line means constant velocity.
Velocity-time graph: area under the graph = distance travelled. Split it into triangles, rectangles and trapezia.
Area of a triangle = ½ × base × height. Area of a trapezium = ½ × (a + b) × h, where a and b are the parallel sides.
Gradient at a point on a curve: draw a tangent there, then use change in y ÷ change in x for two points on the tangent.
Circle, centre (0, 0), radius r: x² + y² = r². A point is on the circle if its coordinates fit the equation.
Tangent at (a, b): the radius has gradient b/a, so the tangent has gradient −a/b.

Common mistake

Reading the number in a circle equation as the radius. In x² + y² = 25 the radius is √25 = 5, not 25.

Worked example

Find the equation of the tangent to the circle x² + y² = 20 at the point (2, 4).

Check the point: 2² + 4² = 4 + 16 = 20, so it is on the circle.
Gradient of the radius from (0, 0) to (2, 4) = 4/2 = 2.
The tangent is perpendicular, so its gradient = −1/2.
y − 4 = −½(x − 2), so y = −½x + 1 + 4 = −½x + 5.

Answer: y = −½x + 5

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