Lesson 1 ~25 min Unit 2 · Non-Linear +85 XP

From Linear to Non-Linear

Move beyond straight lines. Discover why curves describe most real-world relationships and meet the three big families: parabolas, hyperbolas, exponentials.

Today's hook: A ball thrown upward follows a curved path. Why can't a straight line describe its height over time?

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Think First

warm-up

A ball thrown straight up reaches a maximum height, then falls back down. If you plot its height $h$ (in metres) against time $t$ (in seconds), the data points form a smooth curve, not a straight line. Sketch what you think this graph would look like, and explain in one sentence why a straight line cannot describe the ball's motion.

Record your answer in your workbook.

The Big Idea

+5 XP

You already know linear relationships: equations like $y = mx + c$ produce straight lines with a constant gradient. But most things in the real world don't change at a constant rate. Areas grow with the square of length, gravity pulls falling objects faster and faster, populations double in fixed periods. These are non-linear relationships, and their graphs are curves.

A linear graph has a constant gradient — equal steps along $x$ produce equal steps along $y$. A non-linear graph has a changing gradient — the same step along $x$ produces a different step along $y$ depending on where you are. In Year 9, three big curve families matter: parabolas ($y = ax^2$), hyperbolas ($y = k/x$) and exponentials ($y = a^x$).

Linear: constant rate | Non-linear: changing rate

Straight or curved?

Plot any 3 points. If a ruler hits all three, it's linear. If not, non-linear.

Look at the equation

$x^2$, $1/x$ or $2^x$ in the equation $=$ guaranteed curve.

First differences

From a table: equal first differences $=$ linear. Changing differences $=$ non-linear.

What You'll Master

objectives

Know

Linear graphs have form $y = mx + c$ and are straight
Non-linear graphs are curves — gradient changes
Three main families: parabola, hyperbola, exponential

Understand

Why curves arise naturally (squaring, dividing, doubling)
How to detect non-linearity from a table of values
The visual difference between each curve family

Can Do

Classify a relationship as linear or non-linear from equation, table, or graph
Plot $y = x$ and $y = x^2$ on the same axes
Identify which family a curve belongs to by inspection

Words You Need

vocabulary

LinearA relationship whose graph is a straight line; equation form $y = mx + c$.

Non-linearA relationship whose graph is curved; gradient is not constant.

ParabolaThe U-shaped curve produced by $y = ax^2 + bx + c$.

HyperbolaThe two-branch curve produced by $y = k/x$.

ExponentialThe rapidly-growing curve produced by $y = a^x$ (base $a > 0$, $a \neq 1$).

GradientThe steepness of a graph. Constant for lines; variable for curves.

Spot the Trap

heads-up

Wrong: "$y = 2x + 3$ is non-linear because it has an $x$ in it." No — the $x$ is to the power 1 only. Linear.

Right: Look for $x^2$, $x^3$, $1/x$, or $x$ as an exponent. Those make it non-linear.

Wrong: "Joining the points with straight segments shows the curve." No — that gives a broken line, not the true curve.

Right: Plot many points and join them with a smooth curve.

Comparing $y = x$ and $y = x^2$

+5 XP

Build the tables side by side for $x = -3, -2, -1, 0, 1, 2, 3$:

$x$	$y = x$	$y = x^2$
$-3$	$-3$	$9$
$-2$	$-2$	$4$
$-1$	$-1$	$1$
$0$	$0$	$0$
$1$	$1$	$1$
$2$	$2$	$4$
$3$	$3$	$9$

$y = x$ steps by $1, 1, 1, \ldots$ | $y = x^2$ steps by $5, 3, 1, 1, 3, 5$

Equal steps?

$y = x$ first differences are constant ($1$). It's linear.

Changing steps

$y = x^2$ differences shrink then grow. Non-linear.

Symmetry clue

$y = x^2$ gives the same $y$ for $x$ and $-x$. The line $y = x$ does not.

Meet the Three Families

+5 XP

You'll spend the next 19 lessons exploring these three curves. Here's a sneak peek.

Parabola — $y = x^2$. A U-shape, used for trajectories, satellite dishes, bridges. Hyperbola — $y = 1/x$. Two branches, used for inverse proportion, lens equations. Exponential — $y = 2^x$. A steeply rising curve, used for population growth, compound interest, viruses.

Three families — three signature shapes

Parabolas

Symmetric U. Has a single lowest (or highest) point.

Hyperbolas

Two separate branches; never touches axes.

Exponentials

Hugs the $x$-axis on one side, skyrockets on the other.

Watch Me Solve It · 3 examples

Watch Me Solve It · Classify from equation

+15 XP per step

PROBLEM

Classify each as linear or non-linear: (a) $y = 4x - 7$ (b) $y = x^2 + 1$ (c) $y = 5/x$.

1
Part (a)
$y = 4x - 7$ — $x$ is to power 1
No $x^2$, no $1/x$, no exponent. Linear.
2
Part (b)
$y = x^2 + 1$ — contains $x^2$
$x$ squared makes it a parabola. Non-linear.
3
Part (c)
$y = 5/x$ — $x$ is in the denominator
$1/x$ pattern $\Rightarrow$ hyperbola. Non-linear.

Answer(a) Linear (b) Non-linear (c) Non-linear

Watch Me Solve It · Classify from a table

+15 XP per step

PROBLEM

Is this relationship linear? $x: 1, 2, 3, 4, 5$ $y: 2, 5, 10, 17, 26$.

1
Find first differences in $y$
$5 - 2 = 3, \quad 10 - 5 = 5, \quad 17 - 10 = 7, \quad 26 - 17 = 9$
2
Are differences constant?
$3, 5, 7, 9$ — NOT constant
Linear relationships have constant first differences.
3
Identify the pattern
Try $y = x^2 + 1$: $1+1=2 \checkmark, \; 4+1=5 \checkmark, \; 9+1=10 \checkmark$
Matches a parabola. Definitely non-linear.

AnswerNon-linear — first differences change. The rule is $y = x^2 + 1$.

Watch Me Solve It · Plot $y=x$ vs $y=x^2$

+15 XP per step

PROBLEM

For $x = -2, -1, 0, 1, 2$, plot $y = x$ and $y = x^2$ on the same axes. State which is linear and which is curved.

1
Build the tables
$y=x$: $(-2,-2),(-1,-1),(0,0),(1,1),(2,2)$ $y=x^2$: $(-2,4),(-1,1),(0,0),(1,1),(2,4)$
2
Plot $y = x$
All five points fall on a straight line through the origin with gradient $1$.
3
Plot $y = x^2$
The points form a U-shape symmetric about the $y$-axis, passing through $(0,0)$.
$y = x$ is linear; $y = x^2$ is non-linear (a parabola).

Answer$y = x$ is the straight line; $y = x^2$ is the U-shaped curve (parabola).

Common Pitfalls

heads-up

Confusing "has an $x$" with "linear"

Any equation with $x$ in it is not automatically linear. The shape depends on what's done to $x$.

Fix: Linear means $x$ appears only to power 1, not inside a denominator or exponent.

Joining points with straight lines

When you have only 5 points, joining them as a polyline does not give the true curve.

Fix: Plot many points (or use the equation's shape) and draw a smooth curve.

Forgetting negative $x$

Students often only plot positive $x$ values, missing important features (symmetry, branches).

Fix: Always include negative $x$ values in your table when sketching.

Copy Into Your Books

Linear vs Non-Linear

Linear: $y = mx + c$, straight line
Non-linear: anything else, curve
Constant gradient $\Leftrightarrow$ linear

Detection

From equation: spot $x^2$, $1/x$, $a^x$
From table: check first differences
From graph: ruler test

Three Families

Parabola: $y = ax^2$
Hyperbola: $y = k/x$
Exponential: $y = a^x$

Real-world

Trajectory $\to$ parabola
Inverse share $\to$ hyperbola
Population doubling $\to$ exponential

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Linear or Non-Linear

4 problems

Four quick problems. Decide, then reveal the answer.

1 Classify $y = 3x + 8$ as linear or non-linear.
Only $x$ to power 1.Linear
2 Classify $y = 6/x$.
$x$ in the denominator $\Rightarrow$ hyperbola.Non-linear (hyperbola)
3 First differences in $y$ are $2, 2, 2, 2$. Linear or not?
Constant differences $\Rightarrow$ straight line.Linear
4 Which family does $y = 2^x$ belong to?
$x$ is the exponent.Exponential

Complete in your workbook.

Quick Check · 5 questions

Which equation is non-linear?

+10 XP

A table gives first differences $3, 3, 3, 3$ in $y$. What can you conclude?

+10 XP

Which equation produces a hyperbola?

+10 XP

Which is an exponential?

+10 XP

What shape is the graph of $y = x^2$?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Build a table of values for $y = x^2$ using $x = -3, -2, -1, 0, 1, 2, 3$. State two things you notice that show the relationship is non-linear.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Classify each as linear or non-linear, and if non-linear, name the family: (a) $y = 7 - 2x$ (b) $y = 3/x$ (c) $y = x^2 + x$ (d) $y = 5^x$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A ball is dropped and its distance fallen $d$ (m) after $t$ seconds is recorded: $t: 0, 1, 2, 3$ and $d: 0, 5, 20, 45$. Show using first differences that the relationship is non-linear, and explain why this matches the physics that gravity makes objects accelerate.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $y = x^2 - 4$ contains $x^2$, so it is non-linear.

2. A — Constant first differences indicate a linear relationship.

3. B — $y = 1/x$ is a hyperbola.

4. D — $y = 2^x$ has $x$ as the exponent — exponential.

5. B — $y = x^2$ is a U-shaped parabola.

Show Your Working Model Answers

Q6 (3 marks): Table: $(-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)$ [1]. First differences are $-5, -3, -1, 1, 3, 5$ — not constant [1]. Same $y$ for $x$ and $-x$ shows symmetry, not a straight-line behaviour [1].

Q7 (2 marks): (a) Linear [0.5]. (b) Non-linear, hyperbola [0.5]. (c) Non-linear, parabola [0.5]. (d) Non-linear, exponential [0.5].

Q8 (4 marks): First differences: $5, 15, 25$ [1]. Differences are not constant; in fact second differences are constant at $10$, indicating a quadratic relationship [1]. The rule is $d = 5t^2$ [1]. Gravity causes acceleration, so the distance fallen each second increases — matching a parabolic curve rather than a straight line [1].

Stretch Challenge · +25 XP, +10 coins

Second Differences Detective

A table gives $y$-values $2, 7, 16, 29, 46$ for $x = 0, 1, 2, 3, 4$. The first differences are $5, 9, 13, 17$ — not constant. Find the second differences. What do they suggest about the type of relationship?

Reveal solution

Second differences: $9-5=4, \; 13-9=4, \; 17-13=4$ — constant at 4. Constant second differences mean the rule is quadratic. The rule here is $y = 2x^2 + 3x + 2$, so the graph is a parabola.

Quick Review

Linear

$y = mx + c$; constant gradient; straight line

Non-linear

Any curve; gradient changes along the graph

Parabola

$y = ax^2$ U-shape; symmetric

Hyperbola

$y = k/x$; two branches; never touches axes

Exponential

$y = a^x$; hugs axis one side, rockets the other

Detection

Equation, table or graph: look for the signature

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