Linear and Non-linear Relations

A relation connects two quantities, like the number of hours you work and the money you earn. Some relations grow by the same amount every step, and their graphs are straight lines. Others speed up, slow down, or level off, and their graphs curve. Learning to tell them apart helps you pick the right tool and make good predictions.

Key ideas

What a relation is

A relation is a connection between two variables, usually called $x$ and $y$. You can show the same relation in different ways:

• a table of values (pairs of numbers),
• a graph (points on a grid),
• an equation (a rule like $y = 2x + 1$),
• a pattern or a real situation.

Usually $x$ is the independent variable (the one you choose or control, like time) and $y$ is the dependent variable (the one that depends on it, like distance).

Linear relations

A relation is linear when its graph is a straight line. That happens when $y$ changes by the same amount every time $x$ goes up by the same step. We say the relation has a constant rate of change.

A relation whose graph is not a straight line is non-linear. Its rate of change is not constant: the graph gets steeper or flatter as you move along it.

First differences

Here is a quick test using a table. Make sure the $x$-values go up in equal steps (for example, by $1$ each time). Then find the first differences: subtract each $y$-value from the next one.

$$\text{first difference} = \text{next } y - \text{previous } y$$

• If the first differences are all the same, the relation is linear.
• If they are not all the same, the relation is non-linear.

$x$	$y = 2x + 1$	first difference	$y = x^2$	first difference
$0$	$1$		$0$
$1$	$3$	$3 - 1 = 2$	$1$	$1 - 0 = 1$
$2$	$5$	$2$	$4$	$3$
$3$	$7$	$2$	$9$	$5$
$4$	$9$	$2$	$16$	$7$

For $y = 2x + 1$ the first differences are always $2$, so it’s linear. For $y = x^2$ they are $1, 3, 5, 7$, which keep growing, so it’s non-linear.

Spotting linear equations

You can often tell just by looking at the equation.

Equation	Linear?	Why
$y = 2x + 1$	yes	$x$ is only multiplied by a number, and a number is added
$y = \dfrac{x}{4} - 3$	yes	dividing $x$ by a number is the same as multiplying by $\dfrac{1}{4}$
$y = x^2$	no	$x$ is squared
$y = 2^x$	no	$x$ is in the exponent
$y = \dfrac{12}{x}$	no	$x$ is in the denominator

A linear equation can be written as $y = mx + b$, where $m$ and $b$ are numbers. (Your curriculum also writes this as $y = ax + b$. It means exactly the same thing.)

Reading the rate of change from a graph

The shape of a graph tells you how the rate of change behaves.

• Straight line: the rate of change is constant. Each step right gives the same rise (or the same fall).
• Curving upward and getting steeper: the rate of change is increasing. The values grow faster and faster.
• Falling and levelling off: the values go down, but by less and less each step. The rate of change is getting closer to zero.

The same four relations as graphs. Only $y = 2x + 1$ is a straight line. (Each graph has its own scale on the $y$-axis.)

Growing and shrinking patterns

Patterns can be linear or non-linear too.

• A linear growing pattern adds the same amount each time: $4, 7, 10, 13, \ldots$ (add $3$).
• A linear shrinking pattern takes away the same amount each time: $30, 26, 22, 18, \ldots$ (subtract $4$).
• A non-linear pattern changes by different amounts: doubling $1, 2, 4, 8, \ldots$ or halving $64, 32, 16, 8, \ldots$

Making predictions

For a linear relation, keep adding the same first difference, or use the equation. For a non-linear relation, look for the pattern in the table (for example, “each value doubles”) and follow that instead. Never assume a curve will keep going in a straight line.

Worked examples

Example 1: Linear or not, from a table

Decide whether each relation is linear or non-linear.

$x$	$0$	$1$	$2$	$3$	$4$
Relation A: $y$	$5$	$8$	$11$	$14$	$17$
Relation B: $y$	$1$	$3$	$9$	$27$	$81$

Solution. The $x$-values go up by $1$ each time, so we can use first differences.

Relation A: $8 - 5 = 3$, $11 - 8 = 3$, $14 - 11 = 3$, $17 - 14 = 3$. The first differences are all $3$, so A is linear. Its graph is a straight line that rises $3$ for every $1$ step right.

Relation B: $3 - 1 = 2$, $9 - 3 = 6$, $27 - 9 = 18$, $81 - 27 = 54$. The first differences are not the same, so B is non-linear. Notice that each $y$-value is $3$ times the one before, so it grows faster and faster.

Example 2: Linear or not, from an equation

Which of these relations are linear?

$$\text{(a) } y = 4x - 7 \qquad \text{(b) } y = x^2 + 1 \qquad \text{(c) } y = \frac{20}{x} \qquad \text{(d) } y = \frac{x}{2} + 3 \qquad \text{(e) } y = 3^x$$

Solution.

(a) Linear. It has the form $y = mx + b$ with $m = 4$ and $b = -7$.

(b) Non-linear. $x$ is squared.

(c) Non-linear. $x$ is in the denominator. For example, the points $(1, 20)$, $(2, 10)$, $(4, 5)$ don’t lie on a straight line.

(d) Linear. $\dfrac{x}{2}$ is the same as $\dfrac{1}{2}x$, so $m = \dfrac{1}{2}$ and $b = 3$.

(e) Non-linear. $x$ is in the exponent.

Check (c) with first differences: $x = 1, 2, 3, 4$ gives $y = 20, 10, \tfrac{20}{3}, 5$. The first differences are $-10$, then about $-3.33$, then about $-1.67$. Not constant, so non-linear. ✓

Example 3: Two growing patterns

Pattern A is made of tiles: Figure 1 has $1$ tile, Figure 2 has $5$, Figure 3 has $9$, Figure 4 has $13$. Pattern B is a set of squares: Figure $n$ is an $n$ by $n$ square of tiles, so it has $1, 4, 9, 16, \ldots$ tiles.

• (a) Describe how each pattern grows.
• (b) Predict the number of tiles in Figure 10 of each pattern.

Solution.

(a) Pattern A: the first differences are $5 - 1 = 4$, $9 - 5 = 4$, $13 - 9 = 4$. It grows by $4$ tiles every time, so it’s linear, with a constant rate of change of $4$ tiles per figure.

Pattern B: the first differences are $3, 5, 7$. They keep getting bigger, so it’s non-linear, and its rate of change is increasing.

(b) Pattern A: from Figure 1 to Figure 10 is $9$ more steps, and each step adds $4$:

$$1 + 9 \times 4 = 1 + 36 = 37 \text{ tiles}$$

(You can also write the rule $t = 4n - 3$. Check: $4(10) - 3 = 37$. ✓)

Pattern B: Figure 10 is a $10$ by $10$ square, so it has $10^2 = 100$ tiles.

Pattern A is bigger at Figure 2 ($5$ tiles vs $4$). Pattern B catches up at Figure 3 (both have $9$), then pulls ahead from Figure 4 on ($16$ vs $13$) and never looks back. That’s what an increasing rate of change does.

Example 4: Two shrinking patterns

A candle is $24$ cm tall and burns down $3$ cm every hour. A ball is dropped from $160$ cm, and each bounce reaches half the height of the one before.

• (a) Make a table for each one, and decide if it’s linear.
• (b) When will the candle burn out? How high is the ball’s 5th bounce?

Solution.

(a)

Hour	$0$	$1$	$2$	$3$	$4$
Candle height (cm)	$24$	$21$	$18$	$15$	$12$

Bounce	$0$ (drop)	$1$	$2$	$3$	$4$
Ball height (cm)	$160$	$80$	$40$	$20$	$10$

The candle’s first differences are all $-3$, so it’s linear (a constant rate of change of $-3$ cm per hour).

The ball’s first differences are $-80, -40, -20, -10$. They’re not the same, so it’s non-linear. The height still goes down, but by less and less each bounce.

(b) The candle loses $3$ cm per hour from $24$ cm: $24 \div 3 = 8$, so it burns out after $8$ hours.

The ball’s 5th bounce is half of $10$ cm, which is $5$ cm.

Common mistakes

Using first differences when the $x$-values don’t go up evenly. For $y = 2x + 1$ with $x = 0, 1, 3, 4$, the $y$-values are $1, 3, 7, 9$ and the differences are $2, 4, 2$. That looks non-linear, but it isn’t: the jump from $x = 1$ to $x = 3$ is two steps. Check the $x$-column first. If the steps aren’t equal, divide each change in $y$ by the change in $x$.

Thinking “increasing” means “linear”. Lots of relations go up without being straight lines, like $y = x^2$ for $x \ge 0$ and $y = 2^x$. Linear means “goes up (or down) by the same amount each step”.

Calling $y = \dfrac{x}{4}$ non-linear. Dividing $x$ by a number is fine: it’s the same as multiplying by a fraction. It’s dividing a number by $x$, as in $y = \dfrac{4}{x}$, that makes a relation non-linear.

Judging from only two or three points. Over a short stretch, a curve can look almost straight. Use a table with several points, or check the equation.

Subtracting in the wrong order. A first difference is the next value minus the previous one. For a shrinking pattern the first differences should be negative, like $-3$ for the candle.

Predicting a curve with a straight line. If the bounce heights are $160, 80, 40$, the next one is $20$ (half), not $0$ (another drop of $40$). Follow the pattern the table actually shows.

Practice

1. (Warm-up) Find the first differences, and decide if the relation is linear.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$2$	$6$	$10$	$14$	$18$

Solution

The first differences are $6 - 2 = 4$, $10 - 6 = 4$, $14 - 10 = 4$, $18 - 14 = 4$. They’re all the same, so the relation is linear, with a constant rate of change of $4$.

2. (Warm-up) Which of these relations are linear?

• (a) $y = 7 - 2x$
• (b) $y = x^3$
• (c) $y = \dfrac{5}{x}$
• (d) $y = 0.5x + 1$

Solution

(a) Linear. It’s $y = -2x + 7$, so $m = -2$ and $b = 7$.

(b) Non-linear. $x$ is cubed.

(c) Non-linear. $x$ is in the denominator.

(d) Linear, with $m = 0.5$ and $b = 1$.

3. (Warm-up) Is this relation linear? Describe how $y$ changes.

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$60$	$30$	$20$	$15$	$12$

Solution

The first differences are $30 - 60 = -30$, $20 - 30 = -10$, $15 - 20 = -5$, $12 - 15 = -3$. They’re not the same, so the relation is non-linear.

The values of $y$ go down, but by less and less each step, so the graph falls and then levels off. (The rule is $y = \dfrac{60}{x}$.)

4. (Core) A row of squares is made from toothpicks. Figure 1 (one square) uses $4$ toothpicks, Figure 2 uses $7$, Figure 3 uses $10$, and Figure 4 uses $13$.

• (a) Is the pattern linear? Explain.
• (b) Explain why the rule $t = 3n + 1$ works, where $n$ is the figure number.
• (c) How many toothpicks are in Figure 25?
• (d) Which figure uses exactly $100$ toothpicks?

Solution

(a) Yes. The first differences are all $3$: each new square needs $3$ more toothpicks.

(b) Start with $1$ toothpick on the left end. Each square adds $3$ more (top, bottom and right side), so $n$ squares use $3n + 1$. Check: $n = 1$ gives $4$, and $n = 4$ gives $13$. ✓

(c) $t = 3(25) + 1 = 76$ toothpicks.

(d) Solve $3n + 1 = 100$: $3n = 99$, so $n = 33$. Figure 33 uses $100$ toothpicks. Check: $3(33) + 1 = 100$. ✓

5. (Core) Is this relation linear? If so, find its rate of change (the change in $y$ for each increase of $1$ in $x$) and its equation.

$x$	$0$	$2$	$4$	$6$	$8$
$y$	$3$	$7$	$11$	$15$	$19$

Solution

The $x$-values go up by $2$ each time (equal steps), and $y$ goes up by $4$ each time. So the relation is linear.

For each increase of $1$ in $x$, $y$ goes up by $4 \div 2 = 2$. So the rate of change is $2$.

When $x = 0$, $y = 3$, so the equation is $y = 2x + 3$.

Check: $x = 8$ gives $2(8) + 3 = 19$. ✓

6. (Core) Complete a table for $y = 2x + 1$, $y = x^2$ and $y = 2^x$ for $x = 0, 1, 2, 3, 4, 5$. Which relation has the largest value at $x = 5$? Which relations are linear?

Solution

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$y = 2x + 1$	$1$	$3$	$5$	$7$	$9$	$11$
$y = x^2$	$0$	$1$	$4$	$9$	$16$	$25$
$y = 2^x$	$1$	$2$	$4$	$8$	$16$	$32$

At $x = 5$, $y = 2^x$ is largest ($32$). Only $y = 2x + 1$ is linear: its first differences are all $2$.

Notice that $y = x^2$ and $y = 2^x$ are tied at $x = 4$ (both $16$), but after that the doubling pattern pulls ahead.

7. (Core) A phone battery is at $100\%$ and drops $8\%$ every hour while streaming video.

• (a) Is the battery level a linear or non-linear relation of time? Explain.
• (b) Write an equation for the battery level $B$ after $h$ hours.
• (c) After how many hours will the battery reach $20\%$?

Solution

(a) Linear. It drops by the same amount ($8\%$) every hour, so the rate of change is constant: $-8\%$ per hour.

(b) $B = 100 - 8h$.

(c) Solve $100 - 8h = 20$: $8h = 80$, so $h = 10$. The battery reaches $20\%$ after $10$ hours.

Check: $100 - 8(10) = 100 - 80 = 20$. ✓

8. (Challenge) You fold a sheet of paper in half again and again. After $0$ folds there is $1$ layer, after $1$ fold there are $2$ layers, after $2$ folds there are $4$ layers.

• (a) Is the number of layers a linear relation of the number of folds?
• (b) How many layers are there after $6$ folds?
• (c) If the paper is $0.1$ mm thick, how thick would the stack be after $10$ folds (if you could fold it that many times)?

Solution

(a) No. The number of layers doubles each time: $1, 2, 4, 8, \ldots$ The first differences are $1, 2, 4, \ldots$, which aren’t constant, so it’s non-linear. (The rule is $L = 2^f$.)

(b) $2^6 = 64$ layers.

(c) After $10$ folds there are $2^{10} = 1024$ layers, so the stack is

$$1024 \times 0.1 = 102.4 \text{ mm}$$

That’s more than $10$ cm, from one thin sheet! Doubling grows very fast.

9. (Challenge) Jordan made this table and says the relation is non-linear because the first differences are $3, 6, 3, 6$. Is Jordan right? Explain, and find the equation if it is linear.

$x$	$1$	$2$	$4$	$5$	$7$
$y$	$4$	$7$	$13$	$16$	$22$

Solution

Jordan is not right. The $x$-values don’t go up in equal steps (they go up by $1, 2, 1, 2$), so the first differences can’t be compared directly.

Divide each change in $y$ by the change in $x$:

$$\frac{3}{1} = 3, \qquad \frac{6}{2} = 3, \qquad \frac{3}{1} = 3, \qquad \frac{6}{2} = 3$$

The rate of change is always $3$, so the relation is linear. Going back one step from $(1, 4)$ gives $x = 0$, $y = 4 - 3 = 1$, so the equation is $y = 3x + 1$.

Check: $x = 7$ gives $3(7) + 1 = 22$. ✓