Mathematics • Year 9 • Unit 2 • Lesson 1
From Linear to Non-Linear
Build the three-step classify-a-relationship habit: check the equation, check the table, check the graph. Work through one fully-worked example, then a guided one, then eight independent problems graduated from foundation to extension.
1. I do — fully worked example
Read every line. Each step explains why we make the call, not just what the answer is.
Problem. Is the relationship $y = 3x^2 - 4$ linear or non-linear? If non-linear, name the family.
Step 1 — Look at the equation.
Scan for: $x^2$, $x^3$, $1/x$ or $x$ as an exponent. Here we see $x^2$.
Reason: a squared term means $x$ is NOT to the power 1 only, so it cannot be linear.
Step 2 — Rule out "linear".
Linear form is $y = mx + c$ — no powers, no fractions in $x$. $3x^2$ breaks that rule.
Reason: the $-4$ is a constant (fine), but the $3x^2$ pushes us out of $y = mx + c$.
Step 3 — Name the family.
The signature term is $x^2$. That signals a parabola.
Reason: parabolas come from $y = ax^2 + bx + c$. Hyperbolas come from $y = k/x$. Exponentials come from $y = a^x$.
Step 4 — Sanity check with a quick table.
$x = -1, 0, 1, 2$ gives $y = -1, -4, -1, 8$. First differences: $-3, 3, 9$ — NOT constant.
Reason: constant first differences would mean linear. Changing first differences confirm non-linear.
Answer: Non-linear — a parabola.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Is $y = 6/x$ linear or non-linear? If non-linear, name the family.
Step 1 — Look at the equation: the $x$ sits in the __________________ of a fraction.
Step 2 — Rule out "linear": the linear form is $y = $ __________________ . An $x$ in the denominator breaks this form.
Step 3 — Name the family: $y = k/x$ pattern $\Rightarrow$ __________________ .
Step 4 — Quick table check ($x = 1, 2, 3, 6$):
$y$ values: $\_\_\_\_, \_\_\_\_, \_\_\_\_, \_\_\_\_$ — first differences: $\_\_\_\_, \_\_\_\_, \_\_\_\_$ (constant? Yes / No)
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (classify only). The middle two are standard (classify + name the family + tiny check). The last two are extension (table or graph evidence required).
Foundation — classify from the equation
3.1 Classify $y = 4x - 7$ as linear or non-linear. 1 mark
3.2 Classify $y = x^2 + 1$. 1 mark
3.3 Classify $y = 8/x$. 1 mark
3.4 Classify $y = 3^x$. 1 mark
Standard — classify and name the family
3.5 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) $y = 5^x$. 2 marks
3.6 A table gives first differences in $y$ of $2, 2, 2, 2$. State whether the relationship is linear or non-linear and explain in one sentence. 2 marks
Extension — back it up with evidence
3.7 Build a table of values for $y = x^2$ using $x = -3, -2, -1, 0, 1, 2, 3$. Compute the first differences in $y$. Use them to show the relationship is non-linear. 3 marks
3.8 The points $(1, 2), (2, 5), (3, 10), (4, 17), (5, 26)$ are given. Show using first differences that the relationship is non-linear, then test whether the rule $y = x^2 + 1$ matches every point. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $y = 6/x$)
Step 1: $x$ sits in the denominator.
Step 2: linear form is $y = \mathbf{mx + c}$. An $x$ in the denominator breaks this form.
Step 3: $y = k/x$ pattern $\Rightarrow$ hyperbola.
Step 4: $y$ values $= \mathbf{6, 3, 2, 1}$. First differences $= \mathbf{-3, -1, -1}$ — NOT constant. Confirms non-linear.
3.1 — $y = 4x - 7$
$x$ is to the power 1 only, no fractions or exponents. Linear.
3.2 — $y = x^2 + 1$
Contains $x^2$. Non-linear (parabola).
3.3 — $y = 8/x$
$x$ sits in the denominator. Non-linear (hyperbola).
3.4 — $y = 3^x$
$x$ is the exponent. Non-linear (exponential).
3.5 — Classify and name
(a) Linear ($y = mx + c$ form).
(b) Non-linear, hyperbola ($k/x$ form).
(c) Non-linear, parabola ($x^2$ term present).
(d) Non-linear, exponential ($x$ in the exponent).
3.6 — Constant first differences
The relationship is linear. Equal jumps in $y$ for equal jumps in $x$ are the defining feature of a straight line.
3.7 — Table for $y = x^2$
Points: $(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)$.
First differences (left to right): $-5, -3, -1, 1, 3, 5$.
These are NOT constant — in fact they themselves increase by $2$ each time. Because constant first differences are the signature of a straight line, $y = x^2$ must be non-linear. (Bonus: the constant second differences of $2$ confirm a quadratic / parabola.)
3.8 — Match the rule
$y$ values: $2, 5, 10, 17, 26$. First differences: $3, 5, 7, 9$ — not constant, so non-linear.
Test $y = x^2 + 1$: at $x = 1, 2, 3, 4, 5$ we get $2, 5, 10, 17, 26$ — every value matches. So $y = x^2 + 1$ is the rule, and the graph is a parabola.