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

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

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

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

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

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