1. 12. Modelling non-linear relationships
Cambridge University Press
 G K Powers 2013
Study guide Chapter 12

2. Quadratic functions Quadratic function has the equation:
To graph a quadratic function:
Construct a table of values.
Draw a number plane.
Plot the points.
Join the points to make a curve in the shape of a parabola.
HSC Hint – Problems involving quadratic functions may involve maximum or minimum values. Look for the maximum or minimum y value (vertical).
Cambridge University Press
 G K Powers 2013

3. Cubic functions A cubic function has the equation:
To graph a cubic function:
Construct a table of values.
Draw a number plane.
Plot the points.
Join the points to make a curve.
HSC Hint – It is often convenient to use a different scale for the x and y axes.
Cambridge University Press
 G K Powers 2013

4. Exponential functions
An exponential function has the equation:
To graph a exponential function:
Construct a table of values.
Draw a number plane.
Plot the points.
Join the points to make a curve.
HSC Hint – When a > 1 it represents exponential growth. When a < 1 it represents exponential decay.
Cambridge University Press
 G K Powers 2013

5. Hyperbolic functions An hyperbolic function has the equation:
To graph a hyperbolic function:
Construct a table of values.
Draw a number plane.
Plot the points.
Join the points to make a curve.
HSC Hint – A hyperbolic function has two branches (or parts) as division by 0 (x = 0) is undefined.
Cambridge University Press
 G K Powers 2013

6. Algebraic modelling
Algebraic modelling occurs when a practical situation is described mathematically using an algebraic function. For example, the speed of a car (s, in km/h) and time taken (t, in hours) is shown below.
A hyperbolic model describes this situation:
HSC Hint – Algebraic models may have limitations that restrict their use.
Cambridge University Press
 G K Powers 2013

7. Direct variation Write an equation relating the two variables.
y is directly proportional to x then y = kx.
y is directly proportional to square of x then y = kx2.
y is directly proportional to cube of x then y = kx3.
y is directly proportional to square root of x then
Solve the equation for k by substituting values for x and y.
Write the equation with the solution for k (step 2) and solve the problem by substituting a value for either x or y.
HSC Hint – Questions containing the words ‘directly proportional’ or ‘varies directly’ are an indication of a direct variation problem.
Cambridge University Press
 G K Powers 2013

8. Inverse variation Write an equation relating the two variables.
y is inversely proportional to x
so the equation is
Solve the equation for k by substituting values for x and y.
Write the equation with the solution for k (step 2) and solve the problem by substituting a value for either x or y.
HSC Hint – The constant of variation (k) for inverse variation problems is the product of two variables: x and y.
Cambridge University Press
 G K Powers 2013