C. Algebra · Linear and Non-Linear Relations

One Relation, Four Representations: Rate of Change and Initial Value

C3.2 A1About 70 minutes

Learning goals

I can represent a linear relation with a model or diagram, a table of values, a graph and an equation.
I can find the rate of change and the initial value in every representation and explain what they mean.
I can tell a direct variation from a partial variation.
I can decide whether to join the points on a graph (continuous) or not (discrete).

Words to know

initial value
The value of the dependent variable when the independent variable is 0. It is the starting amount, or the fixed part of a linear relation.

direct variation
A linear relation with an initial value of 0, written y = mx. Its graph passes through the origin (0, 0).

partial variation
A linear relation with an initial value that is not 0, written y = mx + b with b ≠ 0. Part of the value is fixed and part changes.

discrete data
Data that can only take separate values (like numbers of tables). Show it with points that are not joined.

continuous data
Data that can take any value in a range (like time or distance). Show it with a solid line or curve.

Planning the winter formal

The student council is setting up square tables for the winter formal. Each table seats one person on each side. When tables are pushed together in a row, the seats where tables touch disappear. How many people fit at a row of 10 tables? Of 25 tables? You could draw them all... or you could find the relation once and use it forever.

The concrete model. Each new table adds 2 seats (one on top, one on the bottom). The 2 end seats are always there, no matter how many tables there are.

The same relation in four ways

1. Words and a model. Each table adds 2 seats, and there are always 2 end seats.

2. A table of values. n = 1, 2, 3, 4 gives S = 4, 6, 8, 10. The first differences are all 2.

3. An equation. Seats = 2 × (number of tables) + 2, or S = 2n + 2.

4. A graph. Plot the points (1, 4), (2, 6), (3, 8), ... They line up because the rate is constant.

The graph of S = 2n + 2. Every step of 1 table to the right goes 2 seats up: that is the rate of change. The open dot at (0, 2) shows the initial value: if you extended the pattern back to 0 tables, only the 2 end seats would be left. The points are not joined because you cannot have 2.5 tables.

Representation	Where to see the rate of change (2)	Where to see the initial value (2)
Model	Each new table adds 2 seats	The 2 end seats that never change
Table of values	The first differences (when n goes up by 1)	The value of S when n = 0 (extend the table backward)
Graph	How far up you go for each 1 to the right	Where the line or pattern of points meets the vertical axis
Equation S = 2n + 2	The number multiplied by n	The number added on (the constant term)

The rate of change and the initial value appear in every representation. Learning to spot them is the key skill of this unit.

Direct and partial variation

Compare two skating rinks. Rink A charges $8 per hour. Rink B charges a $10 entry fee plus $3 per hour.

Rink A: C = 8h. The initial value is 0: skate for 0 hours and you pay nothing. This is a direct variation. Its graph goes through the origin, and doubling the time doubles the cost.
Rink B: C = 3h + 10. The initial value is $10. This is a partial variation: part of the cost is fixed ($10) and part varies ($3 per hour). Doubling the time does not double the cost.

Both relations are linear, so both graphs are straight lines.

Rink A (solid line) is a direct variation: it starts at the origin. Rink B (dashed line) is a partial variation: it starts at $10 on the vertical axis and rises more slowly, $3 for each hour instead of $8. Time is continuous, so each relation is drawn as an unbroken line (the dashes only help you tell the two lines apart). The lines cross at 2 hours, where both rinks cost $16.

Worked example

Example 1: Toothpick squares

A row of squares is made from toothpicks. 1 square uses 4 toothpicks, 2 squares use 7, and 3 squares use 10. Represent the relation with a table, an equation and a graph description, and find the rate of change and initial value.

Table: n = 1, 2, 3, 4 gives T = 4, 7, 10, 13. First differences: 3, 3, 3. Linear.
Rate of change: 3 toothpicks per square. In the model, each new square needs 3 new toothpicks (top, bottom and right side).
Initial value: go back one step from n = 1: 4 − 3 = 1. In the model, this is the very first left-hand toothpick.
Equation: T = 3n + 1. Check: n = 3 gives 3(3) + 1 = 10 ✓.
Graph: points (1, 4), (2, 7), (3, 10), ... rising 3 for every 1 to the right, not joined because the number of squares is discrete.

T = 3n + 1: rate of change 3 toothpicks per square, initial value 1. It is a partial variation.

Worked example

Example 2: From words to an equation, and back

A snow-removal company charges a $40 sign-up fee plus $25 per visit. (a) Write an equation. (b) Make a table for 0 to 4 visits. (c) Find the cost of 12 visits. (d) How many visits cost $290?

(a) The fixed part is $40 (initial value). The rate is $25 per visit. C = 25v + 40.
(b) v = 0, 1, 2, 3, 4 gives C = $40, $65, $90, $115, $140.
(c) C = 25(12) + 40 = 300 + 40 = $340.
(d) Solve 25v + 40 = 290. Subtract 40: 25v = 250. Divide by 25: v = 10 visits.

C = 25v + 40; 12 visits cost $340; $290 buys 10 visits. This is a partial variation.

Worked example

Example 3: Is it direct variation?

Mila earns $16.50 per hour at a part-time job. Is the relation between hours and earnings a direct variation? Show it two ways.

Equation: E = 16.5h. There is no constant added, so the initial value is 0: direct variation.
Table check: h = 2, 4, 8 gives E = $33, $66, $132. Doubling the hours doubles the pay, and E ÷ h = 16.5 every time.
In a partial variation such as C = 3h + 10, the ratio C ÷ h is not constant: 13 ÷ 1 = 13 but 16 ÷ 2 = 8.

Yes, E = 16.5h is a direct variation: it passes through (0, 0) and E ÷ h is always 16.5.

Just for fun

Why are linear relations such reliable friends?

They never change their rate on you.

Hands-on activity

Triangle chains

You need: about 25 toothpicks, cotton swabs or pieces of dry spaghetti, paper, a pencil.

Make one triangle. Then add triangles in a row, each sharing a side with the one before (point up, point down, point up, ...).
Record a table: number of triangles (1 to 6) and number of sticks.
Find the rate of change and the initial value. Point to where each one shows up in your model.
Write an equation, then use it to predict how many sticks 50 triangles need. Check your rule for a small number by building it.

What to look for: 3, 5, 7, 9, 11, 13 sticks. Each new triangle adds 2 sticks, and the initial value is 1 (the first stick), so S = 2t + 1. 50 triangles need 2(50) + 1 = 101 sticks.

Your turn

Use T = 3n + 1 (toothpick squares). How many toothpicks make 20 squares?

Which toothpick-square figure uses exactly 100 toothpicks?

Use S = 2n + 2 (formal tables). How many people fit at 15 tables in a row?

State the rate of change and the initial value of y = 4.5x + 12.

Write an equation for this table.

x: 0, 1, 2, 3

y: 7, 11, 15, 19

Direct or partial variation? (a) C = 12n (b) C = 12n + 5

Describe a situation that d = 80t could model. Is it direct or partial variation?

More practice: warm-up

What is the initial value of y = 3x + 8?

What is the rate of change of y = 3x + 8?

Does the graph of y = 5x pass through the origin?

A table starts at x = 0 with y-values 2, 5, 8, 11. What is the initial value?

Number of tables and number of seats: discrete or continuous?

More practice: on your own

A gym charges a $30 sign-up fee plus $15 per month. Write an equation and find the cost for one year.

x: 0, 1, 2, 3 and y: 50, 45, 40, 35. Write an equation and describe a situation it could model.

A line goes through (0, 6) and (1, 9). Write its equation.

A taxi charges C = 3.5k + 4, where k is kilometres. How far can you go for $25?

Why should the points of S = 2n + 2 not be joined with a solid line, but the points of C = 3h + 10 (skating) should?

Challenge

Hexagons are made in a row from toothpicks, each sharing one side with the one before. 1 hexagon uses 6 toothpicks, 2 use 11, 3 use 16. Write an equation, then find how many hexagons use 201 toothpicks.

Jordan made this table: x = 0, 1, 2, 3 and y = 4, 7, 10, 13. He wrote the equation y = 4x + 3. Find and fix his mistake.

y varies directly with x, and y = 18 when x = 4. Find y when x = 10.

Check your understanding

Quiz

In the equation C = 25v + 40, what is the initial value?

Which relation is a direct variation?

x: 0, 1, 2, 3 and y: 5, 9, 13, 17. Which equation matches?

On a graph, where do you see the initial value?

Toothpick squares use T = 3n + 1. How many toothpicks for 10 squares?

Which situation should be graphed with separate points (not joined)?

For parents and teachers

Big idea: Expectation C3.2 asks students to represent linear relations with concrete materials, tables of values, graphs and equations and to connect them through the rate of change and initial value. These two numbers are the heart of the unit: in the next lessons they become the slope and the y-intercept.

Common difficulties:

Swapping the rate and the initial value when writing an equation from a table. Ask: "What do you add each time?" (rate) and "What do you have at zero?" (initial value).
Reading the initial value from the first row of a table that does not start at 0. Work backwards one step.
Joining points for discrete data. Ask: "Does halfway between make sense?"

How to help: Build the toothpick or triangle patterns together and ask your child to point to the rate and the initial value in the model. Look for direct and partial variations on bills (a fixed monthly fee plus a charge per use).