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.

The radius is any line segment that joins the centre of a circle to its edge.

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

Finding the equation of the tangent to a circle

Maths

Checking and securing understanding of reading from context-based graphs

Checking and securing understanding of drawing distance-time graphs

Distance-time graphs

Speed-time graphs

Non-linear distance-time graphs

Interpreting and drawing real-life graphs

Interpreting and drawing more real-life graphs

Calculating the rate of change

Estimating the gradient of a curve

Improving the estimate of the gradient of a curve

Finding the equation of a radius of a circle

Calculating journeys from linear speed-time graphs

Estimating journeys from non-linear graphs

Efficiently estimating journeys from non-linear graphs

Advanced problem solving with real-life graphs

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.

In this unit pupils expand upon knowledge developed in the linear graphs unit. Pupils are introduced to rates of change and the gradient of the tangent as a way of estimating the gradient of a curve.

Real-life graphs

A tangent of a circle is a line that intersects the circle __________.

The $$x$$-coordinate of the centre is $$7$$

The $$x$$-coordinate of the centre is $$-7$$

The $$y$$-coordinate of the centre is $$10$$

The $$y$$-coordinate of the centre is $$-10$$

From the equation: $$(x-7)^2+(y-10)^2=25$$ we can deduce what?

The equation of a circle is $$(x-2)^2+(y-3)^2=80$$ Which of the below coordinate pairs will be on the radius connecting the centre of the circle to the point $$(6,11)$$ on the circumference?

The gradient of the radius connecting a circle's centre to the point $$(4,11)$$ is $$4$$. What is the gradient of the tangent to the circle at this point?

The gradient of the radius connecting a circle's centre to the point $$(6,6)$$ is $$-{1\over3}$$. What is the equation of the tangent to the circle at this point? Answer in the form $$y=mx+c$$

What is the equation of the tangent to the circle $$(x+15)^2+(y-8)^2=13$$ at the point $$(-13,5)$$?

The tangent at any point on a circle __________ to the radius at that point.

What is the gradient of the line passing through the points $$(-7,-8)$$ and $$(-9,10)$$?

What is the equation of the line passing through the points $$(8,11)$$ and $$(-12,6)$$?

Match the centre of each circle to the equation for the circle.

What is the gradient of the radius connecting the centre of the circle $$(x-175)^2+(y+65)^2=250$$ to the point $$(190,-70)$$ on the circumference?

What is the equation of the radius connecting the centre of the circle $$(x-5)^2+(y+8)^2=10$$ to the point $$(4,-5)$$ on the circumference?

Starter Quiz

The tangent at any point on a circle __________ to the radius at that point.