1. 9. Modelling linear relationships
Cambridge University Press
 G K Powers 2013
Study guide Chapter 9

2. Linear functions A linear function makes a straight line graph.
To graph a linear function follow these steps:
Construct a table of values with the independent variable (x) as the first row and the dependent variable (y) as the second row.
Draw a number plane with the independent variable on the horizontal axis and the dependent variable as the vertical axis. Plot the points.
Join the points to make a straight line.
HSC Hint – Check the points are plotted correctly if the linear function is not a straight line graph.
Cambridge University Press
 G K Powers 2013

3. Gradient-intercept formula
Linear equations in the form
m – Gradient or slope of the line.
b – y-intercept.
Sketching a straight-line requires at least two points. When an equation is written in gradient-intercept form, one point on the graph is immediately available: the y-intercept. A second point can be quickly calculated using the gradient.
HSC Hint – Check the graph by selecting a point on the line and substituting it into the linear equation.
Cambridge University Press
 G K Powers 2013

4. Linear functions as models
Linear modelling occurs when a practical situation is described mathematically using a linear function. For example, the gradient-intercept form of a straight-line graph can sometimes be used to model catering costs. A catering company charges a base amount of $100 plus a rate of $25 per guest. Let the c be the cost of the event ($) and n be the number of guests, we can write
HSC Hint – Linear functions as models are often restricted such as n ≥ 0 and a whole number (see above example).
Cambridge University Press
 G K Powers 2013

5. Intersecting graphs
When the point of intersection of two straight lines is found it is said to be solving the equations simultaneously.
To solve two linear equations simultaneously:
Draw a number plane
Graph both linear equations on the number plane.
Read the point of intersection of the two straight lines.
HSC Hint – The point of intersection of two linear equations satisfies both equations.
Cambridge University Press
 G K Powers 2013

6. Break-even analysis
Break-even point occurs when costs equals income. There is no profit or loss at the break-even point.
Income
Costs
HSC Hint – Profit or loss is calculated by drawing a vertical line and estimating the difference between the income and the costs.
Cambridge University Press
 G K Powers 2013