The gradient is a measure of how steep a line is. It is calculated by finding the rate of change in the $$y$$-direction with respect to the positive $$x$$-direction.

A tangent of a circle is a line that intersects the circle exactly once.

Advanced problem solving with real-life graphs

Maths

The work on gradient in the prior unit is expanded further in this unit by considering gradient as the rate of change and interpreting this in context. The gradient of the tangent is introduced as a way of estimating the gradient of a curve.

Real-life graphs

The __________ to a circle from an external point are equal in length.

Find the point of intersection of the tangents to the circle from points A and B.

Tangents from A and B meet at point C. To show that AC $$=$$ BC we would use __________.

'a' has travelled $$90$$ metres and 'b' has travelled $$120$$ metres

'a' has travelled $$90$$ metres and 'b' has travelled $$135$$ metres

'a' has travelled $$250$$ metres and 'b' has travelled $$275$$ metres

How far have these cars travelled between $$0$$ and $$6$$ seconds?

After how many seconds will car 'a' overtake car 'b'? {{}} seconds.

This speed-time graph shows __________ rate of change between time and speed.

What speed is this vehicle travelling at after $$10$$ seconds? {{}} m/s

What distance has this vehicle travelled after $$10$$ seconds? {{}} m/s

Which equation generalises how far this vehicle has travelled after $$x$$ seconds?

What is the $$y$$-coordinate at the point of intersection of the two linear equations $$y=2x-10$$ and $$y=-{1\over2}x+5$$ ? {{}}

What is the equation of the tangent to the circle at point A?

Starter Quiz