Secondary 2 Relations and Functions Worksheet

2 of the 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 2 of the 7 questions are printed below. The other 5 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, two quantities change together.

   1. A skatepark charges $\$4$ to get in, plus $\$2$ for every hour you rent a board. You are interested in the total cost of a visit.
   2. A $300\text{ L}$ water tank is being emptied at a steady $25\text{ L}/\text{min}$. You are interested in how much water is left.
   3. A hockey team must pay a $\$180$ tournament fee, shared equally by the players who sign up. You are interested in what each player pays.

   For each one, name the independent variable and the dependent variable, then say whether the relation is a linear one or an inverse variation, with one sentence of reason.

2. Q2Finding the Equation of a Linear Function

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

3. Q3Pattern Rule of a Sequence

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

4. Q4The Inverse Variation Function (Inversely Proportional Situation)

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

5. Q5The Modes of Representation of a Relation

   Someone leaves the door of a walk-in freezer open. At that moment the temperature inside is $-4^{\circ }\text{C}$, and it then rises by $2^{\circ }\text{C}$ every hour.

   1. Complete a table of values for $x=0,1,2,3,4,5$ hours.
   2. Plot the points on the grid and join them.
   3. Write the rule of the relation.
   4. After how many hours does the freezer reach $0^{\circ }\text{C}$?

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

6. Q6The 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.

7. Q7Synthesis — drawing on several sheets in this topic

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

The 7 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.

One test tells the two models apart, and it is not "does it go down"

The commonest wrong rule at this level is that a relation where one quantity falls while the other rises must be an inverse variation. A tank draining at a steady rate falls while time rises, and it is perfectly linear. What separates the two models is not the direction but what stays constant.

Finding a and b from a table

The rate of change is a division, and it is the same division wherever you take it from — which is also how you check it:

The y-intercept is what the quantity is worth before anything happens — the delivery fee before any drink, the snow already on the ground when the storm began, the height of the seedling on the day it was planted. Find it by taking any point from the table and undoing the rate: if a = 4 and the pair (3, 14) is in the table, then b = 14 − 4 × 3 = 2. Two of the questions here never give you the value at x = 0, which is the point: it is something you work out, not something you read off.

The rule is a sentence about the situation

y = 4x + 2 says "start at 2, add 4 for every unit of x". Both numbers mean something in the story, and questions ask for that meaning in words: 4 cm of snow per hour, 2 cm already on the ground. Writing the rule and never saying what its two numbers are is a half-answer.

The inverse variation: a constant product, and a rule with x underneath

When a fixed total is shared out — one bus among the passengers, one job among the volunteers — the two quantities multiply to that total. Find the constant by multiplying any pair, and the rule follows:

The behaviour is worth noticing, because it is what students expect least: the drops get smaller and smaller as x grows. Going from one to two halves the value; going from three to four barely changes it. There is no constant "amount per step" for this model, and one question makes you compute three consecutive changes to see it.

Pattern rules: count what each new figure adds

A row of squares sharing toothpicks grows by 3 each time, because the fourth side is already there, and starts from a single extra toothpick — so the rule is y = 3n + 1. Notice the same two numbers doing the same two jobs as in any linear rule: the repeated amount multiplies the variable, and the one-off amount stands alone.

The reason this sheet then asks which totals are possible is that a pattern rule only accepts whole figure numbers. Solving 3n + 1 = 75 gives no whole answer, so no figure uses 75 toothpicks; the possible totals are exactly the numbers one more than a multiple of 3. That is a small piece of real reasoning, and it is the kind of question that separates having found a rule from having understood one.

Four ways of saying the same thing

Words, table, graph, rule. Being fluent means being able to start from any one and produce the other three — and, when two of them disagree, working out which one is lying. That is exactly the shape of one of the harder questions here: three descriptions of one food drive, two of which agree. The majority is not automatically right, but checking a rule against a table takes one substitution, and that settles it.

Getting the most out of it

Do the rate of change twice, from different columns

It costs one extra division and it confirms two things at once: that the relation really is linear, and that you divided by the right step in x. On a table whose x-values jump unevenly, this is the check that saves the question.

Say the rule in words before you write it in symbols

"Five dollars to deliver, then five-fifty a drink." Once the sentence is right, the rule is almost impossible to write backwards — and the two numbers land in the right places without you having to remember which is which.

Multiply across every table you meet

Before deciding what kind of relation you are looking at, take the differences and take the products. One of them will be constant, or neither will. It is a ten-second habit that answers a question students otherwise decide by how the story feels.

Common questions