Now that we have the skills and techniques to construct, draw and analyse linear functions, we will put them into context.
Analysing a graph
When we look at a linear function, one of the first things we notice is that the line extends in both directions indefinitely, with no gaps or holes in our line. This is due to the fact that the linear functions are used to model continuous data, that is, data which represents a measured quantity.
When we look further, we can interpret various features of the graph.
Let's say that we have a linear function of the form $P=mt+c$P=mt+c, where $P$P represents profit, and $t$t represents time in weeks.
- The vertical intercept, in this case $c$c, provides us with information on the initial or starting amount of the variable represented by the vertical axis. In this example, it would tell us the initial profit.
- The gradient, or $m$m value, provides us with information on the rate of change. For our example, for every week that goes by, $m$m would show us the increase or decrease in the amount of profit earned.
- We could also look at the $x$x-intercept of the graph and in our example this would tell us when neither a profit nor loss has been made, and would thus show us the "break-even" point.
Practice questions
question 1
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side.
What is the gradient of the function?
What is the $y$y-value of the $y$y-intercept?
Write an equation to represent the amount of water remaining in the bucket, $y$y, as a function of time, $x$x.
What does the slope tell you?
The amount of water remaining in the bucket after $2$2 minutes.
AThe amount of water that is flowing out of the hole every minute.
BThe time it takes for the bucket to be completely empty.
CThe time it takes the amount of water remaining in the bucket to drop by one litre.
DWhat does the $y$y-intercept tell you?
The capacity of the bucket.
AThe amount of water remaining in the bucket after $30$30 minutes.
BThe amount of water remaining in the bucket when it is empty.
CThe size of the hole.
DFind the amount of water remaining in the bucket after $54$54 minutes.
question 2
A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. After conducting some research, they discover that the cost of employing people to make the clothing is $y=800+60x$y=800+60x, where $y$y is the cost and $x$x is the number of t-shirts to be made, while the cost of using machinery (which includes the cost of purchasing the machines) is $y=3200+20x$y=3200+20x.
Which of the following graphs correctly depicts the two cost functions?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DState the value of $x$x, the number of t-shirts to be produced, at which it will cost the same whether the t-shirts are made by people or by machines.
State the range of values of $x$x, the number of t-shirts to be produced, for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of $x$x, the number of t-shirts to be produced, at which it will be more cost efficient to employ people to manufacture the t-shirts.
Creating a linear function from information
Often we are not given the graph, and instead we are given information to help us form a linear function in the form $y=mx+c$y=mx+c which we can then use to graph and/or analyse the context given.
This information is usually provided in one of two ways:
- We are given a rate and an initial amount, thus providing us with the $m$m value and $c$c value respectively.
- We are given four numbers, two of which correspond to the $x$x variable we are analysing, and two of which correspond to the $y$y variable we are analysing. These can then form two pairs of coordinates, giving us enough information to first calculate $m$m and then find $c$c. This information could be provided in the form of a table.
Practice questions
question 3
A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.
Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.
What is the gradient of the function?
What does this gradient represent?
The total amount charged increases by $\$45$$45 for each additional hour of work.
AThe minimum amount charged by the carpenter.
BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.
CThe total amount charged for $0$0 hours of work.
DWhat is the value of the $y$y-intercept?
What does this $y$y-intercept represent?
Select all that apply.
The total amount charged increases by $\$150$$150 for each additional hour of work.
AThe maximum amount charged by the carpenter.
BThe callout fee.
CThe minimum amount charged by the carpenter.
DFind the total amount charged by the carpenter for $6$6 hours of work.
question 4
A phone salesperson earned $\$1200$$1200 in a particular week during which she sold $22$22 units and $\$1350$$1350 in another week during which she sold $27$27 units.
Let $x$x be the number of units sold and $y$y be the weekly earnings.
Find the linear equation that models the units-earnings relationship for this salesperson.
Use your equation to predict the earnings of the salesperson if she sells $35$35 units.
Applications in geometry
Many geometrical properties of figures can either be verified or proved using coordinate geometry.
Here are some useful formulas:
- The distance formula given by $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=√(x2−x1)2+(y2−y1)2
- The midpoint formula is given by $M=$M= $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(x1+x22,y1+y22)
And recall from our previous lesson:
- Two lines that are parallel have the same gradient.
- Two lines are perpendicular if the product of their respective gradients is equal to $-1$−1.
Practice questions
Question 5
The vertices of $\triangle ABC$△ABC are $A\left(9,-12\right)$A(9,−12), $B\left(4,4\right)$B(4,4) and $C\left(-8,-5\right)$C(−8,−5). The sides $AB$AB and $AC$AC have midpoints $D$D and $E$E respectively.
Find the coordinates of points $D$D and $E$E.
$D=$D=$\left(\editable{},\editable{}\right)$(,)
$E=$E=$\left(\editable{},\editable{}\right)$(,)
Find the gradient of side $BC$BC.
Find the gradient of side $DE$DE.
Are $BC$BC and $DE$DE parallel to each other?
Yes
ANo
B
Question 6
Consider the triangle shown below:
Determine the gradient of the line segment $AB$AB.
Similarly, determine the gradient of side $AC$AC:
Next determine the exact length of the side $AB$AB.
Now determine the exact length of the side $AC$AC.
Hence state the type of triangle that has been graphed. Choose the most correct answer:
An equilateral triangle.
AAn acute isosceles triangle.
BAn isosceles right-angled triangle.
CA scalene right-angled triangle.
D