Secondary 3 Relations and Functions Worksheet

11 of the 15 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 11 of the 15 questions are printed below. The other 4 are built on a diagram or a table of values that does not translate to the page, so they are in the free PDF — marked below where they would have come.

1. Q1Algebra — Relations and Functions

   In each situation below, name the independent variable and the dependent variable, then decide whether the dependent variable is a function of the independent one — that is, whether each value of the independent variable gives exactly one value of the dependent variable.

   1. Amélie is paid $14 for each hour she works at a garden centre. We look at how much she earns.
   2. A guidance counsellor records, for every student in a class, the student's shoe size and the student's height.
   3. A snow-clearing truck travels at a steady $30\text{km}/\text{h}$. We look at the distance it has covered since it left the garage.
2. Q2Finding the Equation of a Linear Function

   A pool is being filled by a hose. After $3$ minutes it holds $46$ litres, and after $8$ minutes it holds $106$ litres. The volume is a linear function of the time.

   1. Find the rate of change and say what it means here.
   2. Find the initial value and write the rule.
   3. How much water is in the pool after $15$ minutes?
3. Q3Graphing a Linear Function

   On $1$ March a snow bank at the corner of a street is $6$ m high. It melts at a steady $1.5$ m per week.

   1. Write the rule for the height $h$, in metres, after $w$ weeks.
   2. Graph the function on the grid for $w$ from $0$ to $4$.
   3. Read from your graph the week in which the snow bank disappears, and check it in the rule.

   A blank Cartesian grid for this question is on the printable PDF.

4. Q4Solving Problems Involving Linear Functions

   A skatepark offers two ways to pay. Option A costs $8 for each visit. Option B costs $50 for a season membership plus $3 for each visit.

   1. Write a rule for the total cost of each option in terms of the number of visits $v$.
   2. For how many visits do the two options cost the same?
   3. Zack expects to go $14$ times this season. Which option should he choose, and how much does he save?
5. Q5The Inverse Variation Function (Inversely Proportional Situation)

   This question is built around a diagram or a table of values. Open it in the PDF.

6. Q6The Inverse of a Function

   This question is built around a diagram or a table of values. Open it in the PDF.

7. Q7The Inverse of the Linear Function

   A photo lab charges $c=0.25n+4$ dollars to print $n$ photos, where the $4 is a fixed handling fee.

   1. Find the rule of the inverse, giving $n$ in terms of $c$.
   2. How many photos can be printed for $19?
   3. Verify your answer in the original rule.
8. Q8The Piecewise Function

   Ma\"eva cycles to her cousin's place. She rides at a steady $20\text{km}/\text{h}$ for the first hour, then rests for half an hour, then rides at $24\text{km}/\text{h}$ for the last half-hour.

   1. How far has she gone after $1$ hour?
   2. What is her rate of change between $1$ h and $1.5$ h, and how would that part of the graph look?
   3. How far has she gone in total after $2$ hours?
9. Q9The Properties of Functions

   This question is built around a diagram or a table of values. Open it in the PDF.

10. Q10The Properties of a Linear Function

    A climbing gym sells a prepaid card worth $240. Each visit takes $12 off the card, so the value left after $n$ visits is $f(n)=-12n+240$.

    1. Give the initial value and the zero of the function, and say what each one means for the card holder.
    2. Is the function increasing or decreasing? Justify using the rate of change.
    3. Give the domain and the range in this context, remembering that a visit cannot be split in half.
11. Q11The Rate of Change (a) and the y-Intercept (b)

    This question is built around a diagram or a table of values. Open it in the PDF.

12. Q12The Role of Parameters in a Linear Function

    Four lines are given by their rules: $$y=2x+3,y=2x-1,y=-2x+3,y=0.5x-1.$$ Answer without drawing any of them, and justify each answer by naming the value in the rule you used.

    1. Which two lines are parallel?
    2. The four lines fall into two groups that cross the vertical axis at the same point. Give each group and the point it crosses at.
    3. Which line falls from left to right?
    4. Of the lines that rise, which rise most steeply?
13. Q13Types of Variables

    For each study, name the independent and the dependent variable, say whether each variable is qualitative or quantitative, and for the quantitative ones say whether they are discrete or continuous.

    1. A snow-clearing crew records the depth of snow that fell, in centimetres, and the time in hours needed to clear one street.
    2. A student council records each student's favourite winter sport and how many students chose it.
    3. A hockey team records the number of players on its roster and the total cost of the jerseys, at $65 each.
14. Q14Zero- and First-Degree Polynomial Functions (Linear)

    For each rule, say whether it is a zero-degree or a first-degree polynomial function, and give its rate of change and its $y$-intercept.

    1. $f(x)=9$
    2. $g(x)=-4x$
    3. $h(x)=6-1.5x$
    4. The cost, in dollars, of $x$ pastries at $2.50 each, with no other charge.
15. Q15Synthesis — drawing on several sheets in this topic

    A diver begins an ascent $30$ m below the surface and rises at a steady $2.5$ m per minute. Depths below the surface are written as negative numbers, so the diver's position is $f(t)=2.5t-30$, where $t$ is the time in minutes.

    1. Give the rate of change and the initial value, and say what each means for the diver.
    2. Find the rule of the inverse, giving the time in terms of the position.
    3. Use the inverse to find when the diver reaches the surface and when the diver passes the $-12$ m mark.
    4. Give the domain and the range of $f$ for this ascent, and say on which interval $f$ is negative.

The 14 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

What makes a relation a function

One test, applied in one direction: each value of the independent variable must give exactly one value of the dependent variable. Q1 gives three situations and asks you to name the two variables before deciding — pay against hours worked, height against shoe size, distance against time — and the middle one is the interesting case, because two students with the same shoe size can perfectly well have different heights.

The linear function: two numbers and everything else follows

A rate of change and an initial value determine the whole thing. Q2 gives two readings from a pool filling and asks you to recover both; Q11 does the same from a table for a rain barrel and insists you say what each number means, with units.

From two points, the rate is that quotient; the initial value is what is left when you put one of the points back into the rule. Both numbers are quantities in the situation, not just coefficients — litres per minute and litres, dollars per gigabyte and dollars — and a question that asks what they mean is asking for those units.

Graphing: the rate of change is a movement, and it has a sign

Q3 asks for a graph of a snow bank melting at a steady rate, on a grid, with the answer then read back off the picture and checked in the rule. The challenge question is a student graphing y = −½x + 3 who makes two separate mistakes in one sentence: he drops the minus sign, and he turns the fraction upside down.

The inverse: swap the two columns, and ask whether it survives

Q6 builds the inverse of a chairlift's altitude table simply by swapping the two rows, and then asks what the inverse tells a skier that the original does not — the original answers "where am I after six minutes", the inverse answers "how long until I am at 700 metres". Q7 does the same algebraically for a photo lab's pricing rule, and Q7's check is the honest one: put your answer through the original rule and see the price come back.

Two things about inverses are worth carrying away from this sheet.

Two situations that are not linear

Q5 shares a fixed rental cost among the members of a band. Doubling the number of members halves what each pays, and the product of the two quantities is the same every time — that constant product is what makes it an inverse variation, and it is the property the challenge question uses to tell a genuine one from an impostor. Two tables both show the second quantity falling as the first rises, and only one of them has a constant product; "goes down as the other goes up" is not a definition.

Q8 follows a cyclist who rides, then rests, then rides faster. Each stretch has its own rate of change — including a rate of zero during the rest, which is a flat piece of graph — and that is a piecewise situation described in ordinary language. The challenge version prices a phone plan that is flat up to a data allowance and then charges per extra gigabyte, and asks you to explain why no single rule of the form c = ag + b can describe it. The answer is that the rate of change is not one number, and a linear rule has only one slot for it.

The properties question, and the description that is nearly right

Q9 reads a January temperature graph for domain, range, initial value, maximum, minimum, the intervals of increase, constancy and decrease, and the times when the temperature is above freezing. The challenge question hands you a written description of the same graph containing three errors, which is much harder than producing the description yourself: you have to check every clause against the picture, including the ones that are correct.

Variables have types, and the types decide the graph

Q13 sorts variables into qualitative and quantitative, and the quantitative ones into discrete and continuous. It is easy to treat this as vocabulary, and the challenge question shows why it is not: a postal code is written with digits and is not a number at all, while a count of customers and a mass of coffee are both numbers that behave completely differently — one can only land on whole values, the other on anything in between. That distinction is what decides whether a graph is a set of separate points or an unbroken line, which is exactly the point Q10 makes about a gym card.

Degree zero is still a function

Q14 puts a constant rule beside first-degree ones. A locker that costs the same whatever happens has a rate of change — it is zero — and its graph is a horizontal line, which is different from having no rate at all. The challenge question asks you to repair the sentence "a zero-degree function has no rate of change", and the repair is one word.

Getting the most out of it

Name the two variables before anything else

Independent and dependent, in words, with units. Nearly every question on this sheet becomes routine once that sentence exists, and several of them are unanswerable until it does — including the ones about whether a relation is a function, which are asked in one direction only.

Write the rule, then read the situation back out of it

Having found the rate of change and the initial value, say what each one is: "the pool gains 12 litres a minute, and it held 10 litres before the hose was turned on." If that sentence sounds wrong, the arithmetic is wrong, and you have caught it before spending a question on it.

Check an inverse by going there and back

Put a number through the original rule, put the result through your inverse, and confirm you land on the number you started with. It settles in ten seconds any doubt about whether you have inverted or merely rearranged something.

Sketch it, even when the question does not ask for a graph

A line through the initial value with the right slope, or a curve falling away in an inverse variation, or a flat stretch where nothing changes. The properties questions are mostly reading, and a picture you drew yourself is much easier to read than a rule.

Common questions