Lesson 15 ~25 min Unit 2 · Non-Linear +85 XP

Comparing Non-Linear Graphs

Four graph families — parabola, circle, hyperbola, exponential. Recognise each from its equation, sketch and key features. One glance at the equation should tell you the shape.

Today's hook: Four equations: $y = x^2$, $x^2 + y^2 = 9$, $y = \dfrac{6}{x}$, $y = 2^x$. Without plotting — which one is the closed curve? Which one never touches the $x$-axis?

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Think First

warm-up

You've now met all four non-linear families: parabolas (Lessons 1–11), circles (12), hyperbolas (13), exponentials (14). Each has a fingerprint — a shape, a key feature, a typical equation. Take these four equations: $y = (x - 1)^2$, $x^2 + y^2 = 16$, $y = \dfrac{4}{x}$, $y = 3^x$. For each, name the family, draw a rough sketch and write ONE feature that the other three don't share.

Record your answer in your workbook.

The Big Idea

+5 XP

One look at the equation should tell you the family — and the family fixes the shape, key features and behaviour. Recognising the family first turns every "sketch the graph" question into a recipe.

Equation $\to$ family $\to$ shape. Squared $x$ only = parabola. $x^2 + y^2$ = circle. $\dfrac{k}{x}$ = hyperbola. Variable in the power ($a^x$) = exponential. Spot the form first; sketch second.

Four families, four fingerprints — learn to spot them instantly.

Equation tells all

Read the form — you don't always need to plot.

Family $\to$ features

Each family has predictable intercepts and asymptotes.

Recognise first

Naming the family halves the work.

What You'll Master

objectives

Know

The four non-linear families and their standard equations
Key features of each: vertex/centre/asymptotes/intercepts
Domain and range patterns for each family

Understand

Why the form of the equation determines the shape
Why circles have no $y = f(x)$ form (they're not functions)
Why exponentials never cross the $x$-axis

Can Do

Identify the family from the equation in one glance
Match an equation to its sketch and vice versa
Compare two graphs using a features table

Words You Need

vocabulary

ParabolaU-shaped curve from $y = ax^2 + \ldots$; has a vertex.

CircleClosed loop $x^2 + y^2 = r^2$; centre and radius.

HyperbolaTwo-branch curve $y = \dfrac{k}{x}$; has two asymptotes.

Exponential$y = a^x$; variable in the power; one horizontal asymptote.

AsymptoteA line the curve approaches but never touches.

SymmetryMirror behaviour about an axis or origin — each family has its own pattern.

Spot the Trap

heads-up

Wrong: Treating $y = x^2$ and $y = 2^x$ as the same family because both have a "power".

Right: In $x^2$, the VARIABLE is the base. In $2^x$, the VARIABLE is the exponent. Totally different shapes.

Wrong: Calling $x^2 + y^2 = 25$ a parabola because it has "$x^2$".

Right: Both $x^2$ AND $y^2$ are present and added — that's a circle. Parabolas have only one squared variable.

The Family Fingerprint Table

+5 XP

Memorise this table — it's the comparison engine for every "identify and sketch" question.

Parabola: $y = ax^2 + bx + c$ — U/inverted U — vertex — symmetric about a vertical axis.
Circle: $x^2 + y^2 = r^2$ — closed loop — centre origin, radius $r$ — not a function.
Hyperbola: $y = \dfrac{k}{x}$ — two branches — asymptotes $x = 0$ and $y = 0$.
Exponential: $y = a^x$ ($a > 0$, $a \neq 1$) — one curve — $y$-int $(0, 1)$ — horizontal asymptote $y = 0$.

$ax^2 \to$ U, $\;x^2+y^2 \to$ loop, $\;\dfrac{k}{x} \to$ 2 branches, $\;a^x \to$ growth.

Squared variable

One $\to$ parabola. Two summed $\to$ circle.

$x$ in denominator

Always a hyperbola — check for $\dfrac{k}{x}$.

$x$ in the power

Exponential — never zero, never negative output (if $a > 0$).

Side-by-Side Key Features

+5 XP

For each family, examiners want the SAME checklist: shape, key point/centre, intercepts, asymptotes (if any), symmetry.

Shape: U / loop / two branches / growth-or-decay curve.
Key point: vertex / centre / NONE (asymptote intersection) / $y$-intercept $(0, 1)$.
Intercepts: parabolas can have 0–2 $x$-ints; circles touch axes at $\pm r$; hyperbolas have NONE; exponentials cross $y$-axis only.
Asymptotes: only hyperbolas and exponentials have them.

Same checklist, four families — compare row by row.

No $x$-int rule

Hyperbolas + exponentials never touch the $x$-axis.

Closed vs open

Only circles are closed loops.

Symmetry test

Parabola: line. Circle: line + point. Hyperbola: origin. Exponential: none.

Watch Me Solve It · 3 examples

Watch Me Solve It · Identify four equations

+15 XP per step

PROBLEM

For each equation, name the family and state one key feature: (a) $y = (x - 3)^2 - 4$, (b) $x^2 + y^2 = 25$, (c) $y = \dfrac{8}{x}$, (d) $y = 2^x$.

1
(a) and (b)
(a) Parabola — vertex $(3, -4)$. (b) Circle — centre $(0, 0)$, radius $5$.
2
(c) and (d)
(c) Hyperbola — asymptotes $x = 0$, $y = 0$; in quadrants 1 and 3 since $k = 8 > 0$. (d) Exponential — $y$-int $(0, 1)$; asymptote $y = 0$.
3
Sanity check
Each equation's form (squared, sum of squares, $\dfrac{k}{x}$, $a^x$) uniquely fixes the family. Sketches all differ.
Reading the FORM first means you don't waste time plotting.

Answerparabola, circle, hyperbola, exponential.

Watch Me Solve It · Match sketches to equations

+15 XP per step

PROBLEM

A sketch shows two branches in quadrants 2 and 4 with the axes as asymptotes. Which equation matches: $y = x^2$, $y = -\dfrac{4}{x}$, $y = 3^x$, or $x^2 + y^2 = 4$?

1
Eliminate by shape
U $\to$ parabola (out). Loop $\to$ circle (out). Growth $\to$ exponential (out). Two branches with axes as asymptotes $\to$ hyperbola.
2
Check the sign of $k$
Quadrants 2 and 4 mean $xy < 0$, so $k < 0$. The hyperbola here is $y = -\dfrac{4}{x}$ ($k = -4$).
3
Confirm
Sub $x = 1$: $y = -4$, point $(1, -4)$ — quadrant 4. Sub $x = -1$: $y = 4$, point $(-1, 4)$ — quadrant 2. Matches.
Shape narrows the family; sign or scale narrows the parameter.

Answer$y = -\dfrac{4}{x}$.

Watch Me Solve It · Compare key features

+15 XP per step

PROBLEM

Compare $y = x^2$ and $y = 2^x$: $y$-intercept, $x$-intercept, behaviour as $x \to -\infty$.

1
$y$-intercepts
$y = x^2$: sub $x = 0$, $y = 0$. So $(0, 0)$. $\;y = 2^x$: sub $x = 0$, $y = 1$. So $(0, 1)$.
2
$x$-intercepts
$y = x^2$: $0 = x^2 \Rightarrow x = 0$ (one). $\;y = 2^x$: $2^x > 0$ always — NO $x$-intercept.
3
Behaviour as $x \to -\infty$
$y = x^2$: $y \to +\infty$ (parabola arm rises on the left). $\;y = 2^x$: $y \to 0^+$ (approaches the $x$-axis from above).
Same input, hugely different output behaviour — the family controls everything.

Answersee step-by-step comparison.

Common Pitfalls

heads-up

Confusing $x^2$ with $2^x$

Parabola vs exponential — same numbers, opposite roles.

Fix: ask "where is the variable?" In $x^2$, variable is the BASE. In $2^x$, variable is the EXPONENT.

Drawing a closed loop for a hyperbola

Two branches accidentally joined through the origin.

Fix: hyperbolas are SEPARATE branches that never meet, never touch the axes.

Forgetting the asymptote on $y = a^x$

Exponential drawn crashing into the $x$-axis on the left.

Fix: $y = a^x > 0$ for all $x$. The curve hugs but never touches $y = 0$.

Copy Into Your Books

Parabola

$y = ax^2 + bx + c$
U or inverted U
Vertex; axis of symmetry
0, 1 or 2 $x$-ints

Circle

$x^2 + y^2 = r^2$
Closed loop
Centre $(0, 0)$, radius $r$
Not a function

Hyperbola

$y = \dfrac{k}{x}$
Two branches
Asymptotes $x = 0$, $y = 0$
$k > 0$: Q1/3, $k < 0$: Q2/4

Exponential

$y = a^x$, $a > 0$, $a \neq 1$
$y$-int $(0, 1)$
Asymptote $y = 0$
$a > 1$ grows, $0 < a < 1$ decays

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Name the Family

4 problems

Four quick problems mixing identification and feature recall.

1 Name the family: $y = \dfrac{-3}{x}$.
$x$ in the denominator.Hyperbola ($k = -3$, branches in Q2 and Q4)
2 Name the family: $x^2 + y^2 = 49$.
Both $x^2$ and $y^2$ summed.Circle, centre $(0, 0)$, radius $7$
3 Name the family: $y = 5^x$. State the $y$-intercept.
Variable in the power.Exponential. $y$-int $(0, 1)$
4 Which families NEVER have an $x$-intercept?
Hyperbolas and exponentials sit fully off the $x$-axis.Hyperbola and exponential

Complete in your workbook.

Quick Check · 5 questions

Which family is $x^2 + y^2 = 16$?

+10 XP

Which family is $y = 3^x$?

+10 XP

Which families have NO $x$-intercept?

+10 XP

The branches of $y = \dfrac{-6}{x}$ lie in:

+10 XP

The $y$-intercept of $y = 7^x$ is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

RecallEasy3 MARKS

Q6. For each equation, name the family and state ONE distinguishing key feature: (a) $y = -(x + 2)^2 + 5$, (b) $x^2 + y^2 = 36$, (c) $y = \dfrac{10}{x}$.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Sketch $y = x^2$ and $y = 2^x$ on the SAME axes for $-2 \le x \le 3$. Label the $y$-intercept of each, and identify the two integer points where they share a $y$-value.

Answer in your workbook.

ReasonHard3 MARKS

Q8. Complete a comparison table for $y = \dfrac{4}{x}$, $y = x^2$ and $x^2 + y^2 = 4$: (a) state the $y$-intercept(s) of each (or write "none"), (b) state the $x$-intercept(s), (c) state which has asymptotes and what they are.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $x^2 + y^2 = 16$ is a circle, radius $4$.

2. A — variable in the power $\Rightarrow$ exponential.

3. B — hyperbola and exponential both avoid the $x$-axis.

4. D — $k = -6 < 0$ puts branches in Q2 and Q4.

5. A — $y$-int is always $(0, 1)$ for $y = a^x$.

Show Your Working Model Answers

Q6 (3 marks): (a) Parabola, vertex $(-2, 5)$, opens down [1]. (b) Circle, centre $(0, 0)$, radius $6$ [1]. (c) Hyperbola, asymptotes $x = 0$ and $y = 0$, branches in Q1 and Q3 [1].

Q7 (3 marks): Table $x = -2, -1, 0, 1, 2, 3$: $x^2 = 4, 1, 0, 1, 4, 9$ [1]. $2^x = 0.25, 0.5, 1, 2, 4, 8$ [1]. Both curves pass through $(2, 4)$ exactly. $y$-ints: $(0, 0)$ for parabola, $(0, 1)$ for exponential. Also share $y = 1$ at $x = -1$ (parabola) and $x = 0$ (exponential) — teacher accepts $(2, 4)$ as the key shared integer point [1].

Q8 (3 marks): (a) Hyperbola: none. Parabola $y = x^2$: $(0, 0)$. Circle: $(0, 2)$ and $(0, -2)$ [1]. (b) Hyperbola: none. Parabola: $(0, 0)$. Circle: $(2, 0)$ and $(-2, 0)$ [1]. (c) Only the hyperbola has asymptotes: $x = 0$ (vertical) and $y = 0$ (horizontal) [1].

Stretch Challenge · +25 XP, +10 coins

Identify the Mystery Graph

A graph has these features: it passes through $(0, 0)$, it has no asymptotes, it has reflective symmetry about a VERTICAL line, and as $x \to \pm \infty$, $y \to +\infty$. (a) Which family must this be? Justify by ruling out the other three. (b) Could the equation be $y = x^2$? What additional information would PIN DOWN the equation uniquely?

Reveal solution

(a) Rule out CIRCLE (closed loop, bounded — can't go to $\infty$). Rule out HYPERBOLA (has asymptotes, two branches). Rule out EXPONENTIAL (no reflective symmetry; goes to $0$ on one side, not $\infty$). So it's a PARABOLA, opening upward. (b) Yes, $y = x^2$ matches all the features. But so does $y = 2x^2$, $y = 5x^2$, etc. To pin it down we need ONE more point on the curve (other than the vertex) — sub it in to solve for $a$.

Quick Review

Parabola

$ax^2 + bx + c$ — U/inverted U

Circle

$x^2 + y^2 = r^2$ — closed loop

Hyperbola

$\dfrac{k}{x}$ — two branches

Exponential

$a^x$ — growth/decay

No $x$-int

Hyperbola, exponential

Closed

Only the circle

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