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-values change by the same amount for constant changes in the 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:

This image shows three linear graphs labeled Increasing, Horizontal, and Decreasing. Ask your teacher for more information.

These graphs are all non-linear:

This image shows 4 nonlinear graphs labeled Quadratic, Cubic, Circle, and Hyperbola. Ask your teacher for more information.

Functions can be identified as linear from a table of values by checking for a constant rate of change in the y-values for a constant change in x.

Here are some examples:

Table of x and y values showing the change in x and y. Ask your teacher for more information.

Constant change in x and in y means that this is a linear relationship.

Constant change in x, but not a constant change in y, means that this is a non-linear relationship.

For a non-constant change in x, non-constant change in y, we can check whether this is linear relationship using the gradient formula to find the gradient m:m=\dfrac{y_2-y_1}{x_2-x_1}

Here, for the first two pairs of x and y-values:m=\dfrac{12-5}{10-3}

So m = 1. Checking the slope for the other values, the gradient is found to be the same. This relationship is linear.

Examples

Example 1

Elizabeth has a gross salary of \$70\,352.43 with an annual income tax deduction of 15\,912.05, loan repayments of \$703.52 per calendar month and a superannuation contribution of 9.5\% of the gross salary.

Calculate her net annual income, rounding your answer to the nearest cent.

Worked Solution

Create a strategy

Subtract the annual tax deduction, superannuation contribution, and annual price of loan repayments from the gross pay

If we want to remove deductions from our gross pay, we can subtract each annual deduction. To find the price of superannuation contribution, multiply the percentage by the gross salary. To find the annual price of loan repayments, multiply the amount contributed in a month by 12.

Apply the idea

The superannuation contribution is 0.095 \times 70\,352.43 = 6683.48085

The annual price of loan repayments is 703.52 \times 12 = 8442.24

So we have:

\displaystyle \text{Net pay}	\displaystyle =	\displaystyle 70\,352.43-15\,912.05-8442.24-6683.48085
	\displaystyle =	\displaystyle \$39\,314.66

Idea summary

A linear relationship is a relationship that has a constant rate of change.

Quadratic function

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 \dfrac{1}{x}, \, x^2 and log(10,x).

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, where b and c can be any number and a can be any number except for zero. For example, the equations y = 2x^2 and y = x^2 - 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.

Examples

Example 2

Consider the equation y = x^{2}.

Complete the following table of values.

x	- 3	- 2	- 1	0	1	2	3
y

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-3:

\displaystyle y	\displaystyle =	\displaystyle x^2
	\displaystyle =	\displaystyle (-3)^2	Substitute -3 to x
	\displaystyle =	\displaystyle 9	Square -3

Similarly, by substituting the remaining x-values into y=x^2, we get:

x	- 3	- 2	- 1	0	1	2	3
y	9	4	1	0	1	4	9

Plot the points in the table of values.

Worked Solution

Create a strategy

Plot the points from the completed table in part (a).

Apply the idea

-5

-4

-3

-2

-1

The ordered pairs of points to be plotted on the coordinate plane are (-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4) and (3,9), which are plotted on the graph.

Plot the curve.

Worked Solution

Create a strategy

Draw a curve through the plotted points on the coordinate plane from part (b).

Apply the idea

-5

-4

-3

-2

-1

Are the y-values ever negative?

Worked Solution

Create a strategy

Use the table of values from part (a) and the graph from part (c).

Apply the idea

From the completed table of values in part (a), we can see that y-values are all positive or 0. Since the graph formed in part (c) is also concave up any other y-values will be further in the positive direction.

So y-values are never negative.

Write down the equation of the axis of symmetry.

Worked Solution

Create a strategy

The x-value of the vertex will be the axis of symmetry.

Apply the idea

The vertex is the minimum or maximum point on the graph. In part (c), the minimum point on the graph is (0,0).

So (0,0) is the vertex of the parabola.

So the equation of the axis of symmetry isx=0

What is the minimum y-value?

Worked Solution

Create a strategy

Get the y-coordinate of the lowest point on the graph.

Apply the idea

In part (c), the lowest point on the graph is (0,0).

So the minimum y-value is y=0.

For every y-value greater than 0, how many corresponding x-values are there?

Worked Solution

Create a strategy

Draw a horizontal line through any point on the curve above the x-axis and find the number of points of intersection.

Apply the idea

-5

-4

-3

-2

-1

In the graph, we can see that the horizontal line on the curve intersects at two points.

So for every y-value greater than 0, there are 2 corresponding x-values. Option C is the correct answer.

Example 3

Consider the equation y = \left(x - 2\right)^{2}.

Complete the following table of values.

x	0	1	2	3	4
y

Worked Solution

Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=0:

\displaystyle y	\displaystyle =	\displaystyle \left(x - 3\right)^{2}	Write the equation
	\displaystyle =	\displaystyle \left(0 - 2\right)^{2}	Substitute 0 to x
	\displaystyle =	\displaystyle \left(-2\right)^{2}	Evaluate the subtraction
	\displaystyle =	\displaystyle 4	Evaluate

Similarly, by substituting the remaining x-values into \left(x - 2\right)^{2}, we get:

x	0	1	2	3	4
y	4	1	0	1	4

Sketch the parabola.

Worked Solution

Create a strategy

Plot the points in the table of values and draw the curve passing through each plotted point.

Apply the idea

-1

The ordered pairs of points to be plotted on the coordinate plane are (0,4),(1,1),(2,0),(-1,1) and (-2,4).

The parabola \left(x - 2\right)^{2} must pass through each of the plotted points.

What is the minimum y-value?

Worked Solution

Create a strategy

Get the y-coordinate of the lowest point on the graph.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the minimum y-value is y=0.

What x-value corresponds to this minimum y-value?

Worked Solution

Create a strategy

Get the x-coordinate of the lowest point on the graph.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the x-value that corresponds to the minimum y-value is x=2.

What are the coordinates of the vertex?

Worked Solution

Create a strategy

The vertex here is the lowest point on the curve.

Apply the idea

In part (b), the lowest point on the graph is (2,0).

So the coordinates of the vertex are (2,0).

Idea summary

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, where b and c can be any number and a can be any number except for zero.

9.01 Non-linear relations

Ideas

Linear and non-linear relationships

Examples

Example 1

Quadratic function

Examples

Example 2

Example 3

Outcomes

U2.AoS3.5

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