Finding Equations of Proportionality and

Plotting Co-ordinates

Slideshow 26, MathematicsMr Richard Sasaki

Room 307

Objectives• Understand how to calculate the

constant of proportionality, k• Be able to construct tables from given

equations for and • Begin to plot graphs for such linear

equations

Constant of ProportionalityWhen , there exists the constant of proportionality, (which can take any real number).If we are given the value of and at a specific point, we can calculate .(If at this point, we know and , we can find .) 𝑦=𝑘𝑥⇒𝑘=¿

𝑦𝑥

We can then substitute the value for into the expression.

𝑘

Building EquationsExampleConsider two variables and where . When . Write an equation for in terms of .As , we can say that (where is the constant of proportionality).When … −8=4 𝑘∴𝑘=−2Now, let’s substitute k = -2 back into our equation.𝑦=−2𝑥

Note: means therefore.

Answers, Part 1

𝑘=2𝑘=4𝑘=

12

𝑘=− 12𝑘=6𝑘=−2

When , for any , when .

𝑦=𝑥3

0≤ 𝑦≤20

Answers, Part 2If 3 , as if . can’t equal 3.

𝑘=2 𝑦=2 𝑥𝑘=−3 𝑦=−3𝑥𝑘=3 𝑦=3 𝑥𝑘=4 𝑦=4 𝑥𝑘=−1 𝑦=−𝑥𝑘=−16 𝑦=−16 𝑥𝑘=− 13 𝑦=− 𝑥3

Building Tables and GraphsWe know how to build tables, but we need to plot that data on graphs. What we need are pairs of .co-ordinates

ExampleBuild a table for and write down the corresponding list of co-ordinates.

0 1 2 30 3 6 9

Co-ordinates are in the form .(𝑥 , 𝑦 )Co-ordinates:(0,0 ) ,(1,3 ) ,(2,6 ) ,(3,9 )

Co-ordinates and GraphsCo-ordinates:(0,0 ) ,(1,3 ) ,(2,6 ) ,(3,9 )We can now plot these on a graph.

-axis-axis

….. (Title)

Values of

Values  of   𝑥

What point is this? (2 ,3)

Gridlines

Plotting Co-ordinatesLet’s plot the points from before!

(0 ,0)(1 ,3)(2 ,6)(3 ,9)(0 ,0)Go right 0 and up 0.(1 ,3)Go right 1 and up 3.(2 ,6)Go right 2 and up 6.(3 ,9)Go right 3 and up 9.

Note: Ignore co-ordinates that don’t fit!

Lastly, draw a line of infinite length.

Answers, Part 1

The graph on the left shows direct proportion.

The line doesn’t pass through the origin on the right.

Answers, Part 2𝑦=4 𝑥 𝑦=−2𝑥

Question 2What does do? Well this question is too big to answer for now…