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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