 G K Powers 2013 Cambridge University Press 5. Interpreting linear relationships Study guide 1

 G K Powers 2013 Cambridge University Press Graphing linear functions  A linear function makes a straight line graph.  To graph a linear function follow these steps: 1. Construct a table of values with the independent variable (x) as the first row and the dependent variable (y) as the second row. 2. Draw a number plane with the independent variable on the horizontal axis and the dependent variable on the vertical axis. Plot the points. 3. 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. 2

 G K Powers 2013 Cambridge University Press Gradient and intercept  Gradient of a line is the slope of the line.  The intercept of a line is where the line cuts the axes. The intercept on the vertical axis is called the y-intercept and denoted by the letter b. The intercept on the horizontal axis is called the x-intercept and denoted by the letter a. HSC Hint – Positive gradients go up to the right (/), negative gradients go down to the right (\). 3

 G K Powers 2013 Cambridge University Press 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. 4

 G K Powers 2013 Cambridge University Press Simultaneous equations  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 1. Draw a number plane. 2. Graph both linear equations on the number plane. 3. Read the point of intersection of the two straight lines. HSC Hint – The point of intersection of two linear equations satisfies both equations. 5

 G K Powers 2013 Cambridge University Press 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). 6