AM2 – Interpreting Linear Relationships Vocabulary: (same as for AM1) algebraic expression axis common difference constant conversion dependent variable equation evaluate expand formulae function gradient independent variable intercept linear simplify solution point of intersection simultaneous linear equations stepwise linear function solve substitute y-intercept

Basic concepts: Generates tables of values from an equation Graphs linear functions, with and without a table of values Calculates gradient and y-intercept and has an understanding of the use of each Identify dependent and independent variables Finds the intersection of a pair of linear equations by graphing Solves problems using simultaneous equations Uses stepwise functions Uses conversion graphs Uses linear equations to model practical situations

Basic Information Unless otherwise stated the horizontal numbers on a number line are the 'x' numbers and the vertical numbers are called the 'y' numbers. So if you are given a table of values and are asked to graph the points, the x value goes first (horizontal) and then the y value (vertical). The point (0,0) is the intersection of the two axes and is called the origin.

Graphing Points from a Table Graph the points from these tables: Graph the line by completing the table of values:

Why is Gradient Important? Carpenters/tilers/builders Companies will only insure their metal or tile roof product against leaking if roofs are built within a certain range of gradients. Why would it cost less to build a roof with a shallower gradient? Why would a building with a shallower roof be more prone to leaking? Dirt bike riders/snow skiers Rather than going straight down a very steep slope, how do bike riders and snow skiers tackle the steep slopes? Why would they do this? Vehicle manufacturers What aspects of a car would be important to test on slopes of varying gradients?

Gradient To find the slope or gradient of a line, we look at how far upwards between two points (rise), and how far across between the same two points (run) gradient = rise run e.g. graph the line joining the points (3,5) and (4,7) = ___ Where a graph cuts the y-axis is called the y-intercept, and this represents the y value when x = 0.

Step it up…

Using the y-intercept Any linear function can be written in the form y = mx + b , where m = gradient and b = y-intercept.

Desmos linear activity

Independent and Dependent Variables Imagine a car travelling at a constant speed of 60 km/h. The graph at right compares the distance travelled with time. This graph can be given by the relation D = 60t. This relation is an example of a function. A function is a rule with two variables. In the above example time, t, is the independent variable. This is the variable for which we can substitute any value. Distance, D, is the dependent variable as its value depends on the value substituted for t. A linear function is a graph that, when drawn, is represented by a straight line. Linear functions are drawn from a table of values. The independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

Using Linear Functions

What is the gradient in this graph and what does it represent?

Simultaneous Equations Consider the problem below. A class has 30 students. There are twice as many girls as boys. How many boys and girls are in this class? We solve this problem by modelling two linear relationships. We can say that G = 30 − B and G = 2B, where G represents the number of girls and B represents the number of boys. The solution to the problem will be the point of intersection of these linear relationships. The point of intersection on these lines is (10, 20). Therefore the solution to this problem is 10 boys and 20 girls. Desmos linear activity

Step or Stepwise Graphs A step graph consists of flat sections with an open circle at one end and a closed circle at the other end. The open circle means that the point is not included in that interval and the closed circle means that the point is included in the interval.