Linear and non-linear relationships
A linear relationship is a relationship that has a constant rate of change. This means that the gradient is a constant value and that the $y$y-values change by the same amount for constant changes in the $x$x-values.
Linear relationships, when graphed, are straight lines. This means that any graph that is not a straight line represents a non-linear relationship.
These graphs are all linear:
These graphs are all non-linear:
Table of values
Functions can be identified as linear from a table of values by checking for a constant rate of change in the $y$y-values for a constant change in $x$x.
Here are some examples:
Constant change in $x$x and in $y$y means that this is a linear relationship.
Constant change in $x$x, but not a constant change in $y$y, means that this is a non-linear relationship.
For a non-constant change in $x$x, non-constant change in $y$y, we can check whether this is linear relationship using the gradient formula to find the gradient $m$m:
$m=\frac{y_2-y_1}{x_2-x_1}$m=y2−y1x2−x1
Here, for the first two pairs of $x$x and $y$y-values:
$m=\frac{12-5}{10-3}$m=12−510−3
So $m=1$m=1. Checking the slope for the other values, the gradient is found to be the same. This relationship is linear.
Practice question
QUESTION 1
Elizabeth has a gross salary of $\$70352.43$$70352.43 with an annual income tax deduction of $\$15912.05$$15912.05, loan repayments of $\$703.52$$703.52 per calendar month and a superannuation contribution of $9.5%$9.5% of the gross salary.
Calculate her net annual income, rounding your answer to the nearest cent.
Non-linear functions
There are a number of non-linear functions which will be particularly useful when investigating data and relationships in the following chapters. In particular, the functions $\frac{1}{x}$1x, $x^2$x2 and $\log_{10}x$log10x.
The quadratic function
Recall that the graph of a quadratic function is called a parabola and that a quadratic function has an equation of the standard form $y=ax^2+bx+c$y=ax2+bx+c, where $b$b and $c$c can be any number and $a$a can be any number except for zero. For example, the equations $y=2x^2$y=2x2 and $y=x^2-3x+4$y=x2−3x+4 are both quadratic.
It is possible to create a graph of a function by generating a table of values and evaluating the function for certain values in its domain. This can be done for quadratic functions by connecting the points in a smooth curve that looks like a parabola.
Practice questions
Question 2
Consider the function $y=x^2$y=x2
Complete the following table of values.
$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Plot the points in the table of values.
Loading Graph...Hence plot the curve.
Loading Graph...Are the $y$y values ever negative?
No
AYes
BWrite down the equation of the axis of symmetry.
What is the minimum $y$y value?
For every $y$y value greater than $0$0, how many corresponding $x$x values are there?
$3$3
A$1$1
B$2$2
C
Question 3
Consider the function $y=\left(x-2\right)^2$y=(x−2)2
Complete the following table of values.
$x$x $0$0 $1$1 $2$2 $3$3 $4$4 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Sketch a graph of the function.
Loading Graph...What is the minimum $y$y value?
What $x$x value corresponds to this minimum $y$y value?
What are the coordinates of the vertex? Give your answer in the form $\left(a,b\right)$(a,b).