Gradient of a line Recap Gradient is a measure of steepness. The gradient of this line is 1. Gradient = Δ𝑦 Δ𝑥

What is the gradient of this line? How do you know?

What is the gradient of this line? How do you know?

What is the gradient of this line? How do you know?

What is the gradient of this line? How do you know?

What is the gradient of this line? How do you know?

x : y = 2 : 1 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. A table of values can help draw a graph Complete the table: 𝒙 1 2 3 4 𝒚 Demonstrate The 𝑥 values must be double the 𝑦 values. The 𝑦 values must be half of the 𝑥 values.

x : y = 2 : 1 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. 𝒙 1 2 3 4 𝒚 0.5 1.5 Demonstrate

Remember a table of values represents a set of coordinates. 𝒙 2 4 𝒚 1 (0,0) (2,1) (4,2) Remember a table of values represents a set of coordinates.

x : y = 2 : 1 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. The y-intercept is 0. Let’s calculate the gradient of the line…

x : y = 2 : 1 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. Gradient = Δ𝑦 Δ𝑥 Gradient = 3 6 Gradient = 2 4 Gradient = 1 2

So the equation of the graph is… x : y = 2 : 1 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. The y-intercept is 0. The gradient is 1 2 So the equation of the graph is… 𝑦= 1 2 𝑥 𝑦=𝑚𝑥+𝑐 𝑦= 1 2 𝑥+0

Your turn. On your whiteboards.. x : y = 1 : 2 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph.

x : y = 1 : 2 (a) Draw the graph of y as a function of x. (b) Write down the equation of your graph. Your table of values could look like this… 𝒙 1 2 3 𝒚 Possible Scaffolding

𝒙 1 2 3 𝒚 4 6 𝑦=2𝑥 Answer

In your books… 𝑥 :𝑦=1:4 𝑥 :𝑦=2:4 𝑥 :𝑦=3:4 𝑥 :𝑦=4:4 𝑥:𝑦=1:5 Draw the graph of 𝑦 as a function of 𝑥 for each ratio 𝑥 :𝑦=1:4 𝑥 :𝑦=2:4 𝑥 :𝑦=3:4 𝑥 :𝑦=4:4 𝑥:𝑦=1:5 Work out the equation for each of your graphs.

If you are stuck this table could help you start: Draw the graph of 𝑦 as a function of 𝑥 for each ratio 𝑥 :𝑦=1:4 𝒙 1 2 3 𝒚 4

Mark your work.. 𝑥:𝑦=1:4 ⟹𝑦= 4 1 𝑥 𝑥 :𝑦=2:4 ⟹ 𝑦= 4 2 𝑥 𝑥 :𝑦=2:4 ⟹ 𝑦= 4 2 𝑥 𝑥 :𝑦=3:4 ⟹𝑦= 4 3 𝑥 𝑥 :𝑦=4:4 ⟹ 𝑦= 4 4 𝑥 𝑥:𝑦=1:5⟹𝑦= 5 1 𝑥

More consolidation… 𝑥 :𝑦=1:3 𝑥 :𝑦=3:1 𝑥 :𝑦=2:3 𝑥 :𝑦=3:2 Draw the graph of 𝑦 as a function of 𝑥 for each ratio 𝑥 :𝑦=1:3 𝑥 :𝑦=3:1 𝑥 :𝑦=2:3 𝑥 :𝑦=3:2 Work out the equation for each of your graphs. y = 3x y = 1/3x y = 3/2x y = 2/3x

Any three coordinates… Mark your work 𝑥 :𝑦=1:3 Any three coordinates… 𝒙 1 2 3 𝒚 6 9 y = 3x y = 1/3x y = 3/2x y = 2/3x 𝑦=3𝑥

(b) 𝑥 :𝑦=3:1 Any three coordinates… Mark your work (b) 𝑥 :𝑦=3:1 Any three coordinates… 𝒙 1 2 3 4 5 6 𝒚 𝟏 𝟑 𝟐 𝟑 𝟒 𝟑 𝟓 𝟑 y = 3x y = 1/3x y = 3/2x y = 2/3x 𝑦= 1 3 𝑥

𝑐 𝑥 :𝑦=2:3 Any three coordinates… Mark your work 𝑐 𝑥 :𝑦=2:3 Any three coordinates… 𝒙 1 2 3 4 5 6 𝒚 1.5 4.5 7.5 9 y = 3/2x y = 2/3x 𝑦= 3 2 𝑥

𝑑 𝑥 :𝑦=3:2 Any three coordinates… Mark your work 𝑑 𝑥 :𝑦=3:2 Any three coordinates… 𝒙 1 2 3 4 5 6 𝒚 𝟐 𝟑 𝟒 𝟑 𝟖 𝟑 𝟏𝟎 𝟑 y = 3/2x y = 2/3x 𝑦= 2 3 𝑥

Challenge The line 𝑦= 2 5 𝑥 passes through the point with coordinates (𝑎,𝑏) where 𝑎,𝑏 ≠0 Write down the ratio 𝑎:𝑏 in its simplest form

In general 𝐼𝑓 𝑥:𝑦=𝑎:𝑏 Write as many different possible linear equations as you can.

In general 𝐼𝑓 𝑥:𝑦=𝑎:𝑏 𝑦= 𝑏 𝑎 𝑥 𝑥 𝑦 = 𝑎 𝑏 𝑏𝑥=𝑎𝑦 𝑥= 𝑎 𝑏 𝑦 𝑦 𝑥 = 𝑏 𝑎 There are more… 𝑥= 𝑎 𝑏 𝑦 𝑦 𝑥 = 𝑏 𝑎