Now that we have the skills and techniques to construct, draw and analyse linear functions, we will put them into context.
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 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.
The amount of water that is flowing out of the hole every minute.
The time it takes for the bucket to be completely empty.
The time it takes the amount of water remaining in the bucket to drop by one litre.
What does the $y$y-intercept tell you?
The capacity of the bucket.
The amount of water remaining in the bucket after $30$30 minutes.
The amount of water remaining in the bucket when it is empty.
The size of the hole.
Find the amount of water remaining in the bucket after $54$54 minutes.
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?
State 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.
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:
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$$45 for each additional hour of work.
The minimum amount charged by the carpenter.
The total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.
The total amount charged for $0$0 hours of work.
What 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$$150 for each additional hour of work.
The maximum amount charged by the carpenter.
The callout fee.
The minimum amount charged by the carpenter.
Find the total amount charged by the carpenter for $6$6 hours of work.
A phone salesperson earned $\$1200$$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.
Many geometrical properties of figures can either be verified or proved using coordinate geometry.
Here are some useful formulas:
And recall from our previous lesson:
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
No
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.
An acute isosceles triangle.
An isosceles right-angled triangle.
A scalene right-angled triangle.