Lesson 16 ~25 min Unit 2 · Non-Linear +85 XP

Graphing Non-Linear Relationships

Systematic graphing: identify the family, build a table of values, choose a sensible scale, plot points, connect smoothly. One reliable method for parabolas, hyperbolas and exponentials.

Today's hook: You're asked to graph $y = \dfrac{12}{x}$ on a grid that goes from $-5$ to $5$ on each axis. Where do the extreme values of $y$ go? Should every grid square be the same size?

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Think First

warm-up

Graphing a non-linear curve from scratch needs four decisions: which $x$-values to choose, what scale to use, what shape to expect, how smoothly to join the points. Take $y = x^2 - 4$. Which integer $x$-values would you pick? What's the $y$-range you'd need to show?

Record your answer in your workbook.

The Big Idea

+5 XP

Every non-linear graph follows the same five-step recipe: identify family $\to$ choose $x$-values $\to$ build a table $\to$ choose scale $\to$ plot and join smoothly. Same recipe, different shape.

The table of values is the bridge from equation to grid. Pick clever $x$-values: symmetric about any vertex or asymptote, including $0$ where possible, both sides of every interesting feature. Then plot and join with a smooth curve — never straight segments.

Family $\to$ $x$-values $\to$ table $\to$ scale $\to$ smooth plot.

Always include $0$

$x = 0$ gives the $y$-intercept for free.

Both sides

Pick negative AND positive $x$-values around the feature.

Five or more points

Three points isn't enough for a smooth curve.

What You'll Master

objectives

Know

The 5-step graphing recipe applied to all non-linear families
How to choose $x$-values for parabolas, hyperbolas and exponentials
How to choose a sensible scale based on the $y$-range

Understand

Why hyperbolas need values close to AND far from the asymptote
Why exponentials need both small AND large $x$ to show growth
Why a parabola needs values symmetric about its axis

Can Do

Build a tidy table of values from any non-linear equation
Plot the points on a number plane with appropriate scale
Join points with a smooth, family-appropriate curve

Words You Need

vocabulary

Table of valuesRows of $x$-values with their $y$-values from the equation; the input to plotting.

ScaleThe number-of-units-per-grid-square. Can differ on $x$- and $y$-axes.

Number planeCartesian grid with $x$ horizontal and $y$ vertical; quadrants 1–4.

PlotMark a single point $(x, y)$ from the table on the grid.

Smooth curveA flowing line through the points — no kinks, no straight bits.

Domain restrictionHyperbolas: $x \neq 0$. Skip $x = 0$ in the table.

Spot the Trap

heads-up

Wrong: Using only $x = 1, 2, 3$ to plot $y = x^2$ — only get half the parabola.

Right: Use $x = -3, -2, -1, 0, 1, 2, 3$ so both arms appear.

Wrong: Substituting $x = 0$ into $y = \dfrac{6}{x}$ and writing $y = 0$ or $y = \infty$.

Right: $x = 0$ is excluded from the domain. Write "undefined" or skip the row entirely.

The 5-Step Graphing Recipe

+5 XP

Use these five steps in this order for every non-linear graph:

Identify the family (parabola / hyperbola / exponential / circle) from the equation form.
Choose $x$-values based on the family (see below).
Build the table: compute $y$ for each $x$. Use exact fractions where possible.
Choose the scale: look at the $y$-range; pick units-per-square so the curve fits.
Plot and join smoothly: dot each $(x, y)$, then connect with a flowing curve.

Always label axes, key intercepts and asymptotes.

Family $\to x$-values $\to$ table $\to$ scale $\to$ plot.

Family first

Tells you which $x$-values matter.

Scale matters

$x$- and $y$-scales can differ — pick to fit the page.

Label everything

Axes, intercepts, equation, scale.

Choosing $x$-Values by Family

+5 XP

Different families need different $x$-value strategies:

Parabola: 5–7 integer values symmetric about the axis (e.g. $x = -3, -2, -1, 0, 1, 2, 3$).
Hyperbola $y = \dfrac{k}{x}$: use $x = \pm 1, \pm 2, \pm 3, \pm 6$ — small and large, both sides — SKIP $x = 0$.
Exponential $y = a^x$: use $x = -2, -1, 0, 1, 2, 3$ — negatives show the asymptote, positives show growth.
Circle: can't use $y = f(x)$ form — instead use compass at centre with radius $r$, or solve $y = \pm\sqrt{r^2 - x^2}$.

Strategy by family — pick $x$ to capture the key behaviour.

Parabola: symmetric

Mirror values across the axis.

Hyperbola: factors of $k$

Whole-number $y$-values make plotting easy.

Exponential: small + big

Show asymptote AND growth in one picture.

Watch Me Solve It · 3 examples

Watch Me Solve It · Graph $y = x^2 - 4$

+15 XP per step

PROBLEM

Graph $y = x^2 - 4$ using a table of values for $x \in \{-3, -2, -1, 0, 1, 2, 3\}$. Choose a sensible scale and label intercepts.

1
Identify and table
Family: parabola, vertex $(0, -4)$, opens up. Table: $x = -3, -2, -1, 0, 1, 2, 3$ gives $y = 5, 0, -3, -4, -3, 0, 5$.
2
Choose scale
$x$ range $-3$ to $3$: $1$ unit per square. $y$ range $-4$ to $5$: $1$ unit per square works. Both scales match here.
3
Plot and join
Plot all $7$ points. Join with a smooth U-shape through them. Label $y$-int $(0, -4)$, $x$-ints $(-2, 0)$ and $(2, 0)$.
Seven points with symmetry — the curve is well constrained.

AnswerU through $(-3,5), (-2,0), (-1,-3), (0,-4), (1,-3), (2,0), (3,5)$.

Watch Me Solve It · Graph $y = \dfrac{6}{x}$

+15 XP per step

PROBLEM

Graph $y = \dfrac{6}{x}$ for $-6 \le x \le 6$, $x \neq 0$, using a table of values with factors of $6$.

1
Choose $x$-values and build table
$x = -6, -3, -2, -1, 1, 2, 3, 6$ (factors of $6$). Then $y = -1, -2, -3, -6, 6, 3, 2, 1$. Skip $x = 0$ (undefined).
2
Scale and plot
$x$ range $-6$ to $6$, $y$ range $-6$ to $6$: $1$ unit per square fits a standard grid. Plot all eight points.
3
Join smoothly — TWO branches
Connect the four points in quadrant 1 with one smooth branch; the four in quadrant 3 with another. NEVER cross the axes.
$k = 6 > 0$ $\Rightarrow$ branches in Q1 and Q3, asymptotes $x = 0$, $y = 0$.

AnswerTwo branches in Q1 and Q3 with axes as asymptotes.

Watch Me Solve It · Graph $y = 2^x$

+15 XP per step

PROBLEM

Graph $y = 2^x$ for $x = -3, -2, -1, 0, 1, 2, 3$. Choose a sensible $y$-scale and label the asymptote.

1
Build table
$y = 2^{-3}, 2^{-2}, 2^{-1}, 2^0, 2^1, 2^2, 2^3 = \tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{2}, 1, 2, 4, 8$.
2
Choose scales
$x$: $1$ unit per square. $y$: $1$ unit per square (max $y = 8$ fits well). For neg $x$, $y$ is fractional — tiny near the axis but never touches.
3
Plot and asymptote
Plot all $7$ points. Smooth curve rising steeply on the right. Left arm hugs the $x$-axis but never touches — label $y = 0$ as the horizontal asymptote.
Asymptote labelling is required — the visual look alone isn't enough.

AnswerGrowth curve through $(0,1), (1,2), (2,4), (3,8)$ with asymptote $y = 0$.

Common Pitfalls

heads-up

Too few points

Plotting only $3$ points then guessing a smooth curve.

Fix: use at least $5$–$7$ points, including values either side of any feature (vertex / asymptote).

Straight-line joins

Connecting plotted points with straight line segments.

Fix: non-linear means non-line. Use smooth curves — no kinks at the plotted points.

Wrong scale crushes the graph

Using $1$ unit per square for $y$ when $y$ reaches $100$ — off the page.

Fix: check the table's $y$-range FIRST, then pick the scale so the highest and lowest $y$ fit comfortably.

Copy Into Your Books

5-step recipe

Identify family
Choose $x$-values
Build table
Choose scale
Plot & join smoothly

$x$-values by family

Parabola: symmetric integers
Hyperbola: factors of $k$
Exponential: $-2$ to $3$
Circle: compass & radius

Scale tips

Check $y$-range first
$x$ and $y$ can differ
Aim to fill the grid

Plotting rules

Smooth, never straight
Label intercepts
Label asymptotes
State the equation

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Build & Plot

4 problems

Four quick problems building tables and choosing scales.

1 Complete the table for $y = x^2 + 1$: $x = -2, -1, 0, 1, 2$.
$y = (-2)^2 + 1, (-1)^2 + 1, 0+1, 1+1, 4+1$.$y = 5, 2, 1, 2, 5$
2 List the best $x$-values for $y = \dfrac{8}{x}$ from $-8$ to $8$.
Factors of $8$, both signs, exclude $0$.$x = \pm 1, \pm 2, \pm 4, \pm 8$
3 Complete the table for $y = 3^x$ at $x = -1, 0, 1, 2$.
$y = 3^{-1}, 3^0, 3^1, 3^2$.$y = \tfrac{1}{3}, 1, 3, 9$
4 The table for $y = 4^x$ gives $y$-values up to $64$. Suggest a $y$-scale (units per square).
$64$ on a standard grid needs big squares per unit.$10$ units per square (so $7$ squares fits $0$–$70$)

Complete in your workbook.

Quick Check · 5 questions

For $y = x^2 - 3$, when $x = -2$, $y = $:

+10 XP

For the table of $y = \dfrac{12}{x}$, which $x$-value must be SKIPPED?

+10 XP

For $y = 2^x$, when $x = -2$, $y = $:

+10 XP

For a parabola $y = x^2 + c$, the BEST set of $x$-values for a table is:

+10 XP

After plotting the table for $y = x^2$, you join the points with:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Build a table of values for $y = x^2 - 2x$ at $x = -1, 0, 1, 2, 3$. Then state the vertex by inspection.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Build a table of values for $y = \dfrac{4}{x}$ at $x = -4, -2, -1, 1, 2, 4$. Sketch the graph and label the asymptotes.

Answer in your workbook.

ReasonHard3 MARKS

Q8. A student wants to graph $y = 3^x$ for $x = -3$ to $x = 4$. (a) Build the table. (b) The $y$-values range from $\tfrac{1}{27}$ to $81$ — explain why a $y$-scale of "$1$ unit per square" is unsuitable, and suggest a better one. (c) State the equation of the asymptote.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $(-2)^2 - 3 = 1$.

2. D — $\dfrac{12}{0}$ is undefined.

3. A — $2^{-2} = \dfrac{1}{4}$.

4. C — symmetric integers capture both arms.

5. B — always smooth curve, never straight.

Show Your Working Model Answers

Q6 (3 marks): $y = 3, 0, -1, 0, 3$ at $x = -1, 0, 1, 2, 3$ [1]. Minimum $y$-value in the table is $-1$ at $x = 1$ [1]. So vertex appears to be $(1, -1)$ [1].

Q7 (3 marks): $y = -1, -2, -4, 4, 2, 1$ at $x = -4, -2, -1, 1, 2, 4$ [1]. Plot $6$ points; smooth Q1 branch and Q3 branch [1]. Asymptotes: $x = 0$ (vertical) and $y = 0$ (horizontal) [1].

Q8 (3 marks): (a) $y = \tfrac{1}{27}, \tfrac{1}{9}, \tfrac{1}{3}, 1, 3, 9, 27, 81$ at $x = -3, -2, -1, 0, 1, 2, 3, 4$ [1]. (b) $1$ unit per square would need $81$ squares vertically — off the page. Use $\sim 10$ units per square (about $9$ squares fits $0$–$90$) [1]. (c) Asymptote $y = 0$ [1].

Stretch Challenge · +25 XP, +10 coins

Mixed Graphing on One Set of Axes

On the same set of axes, you must plot $y = x^2$, $y = \dfrac{6}{x}$ (positive branch only), and $y = 2^x$ for $x = 1, 2, 3$. (a) Build the table for all three. (b) Which curve passes through the highest point at $x = 3$? (c) At $x = 2$, list the three points in order from lowest $y$ to highest $y$. (d) Suggest a $y$-scale that fits all three curves on the same grid.

Reveal solution

(a) $x = 1$: $1, 6, 2$. $\;x = 2$: $4, 3, 4$. $\;x = 3$: $9, 2, 8$. (b) At $x = 3$: parabola $y = 9$ is highest (just above exponential $y = 8$, hyperbola $y = 2$). (c) At $x = 2$: hyperbola $3$ < parabola $4 =$ exponential $4$ (tie). So order: $\dfrac{6}{x} (3)$, $x^2 = 2^x (4)$. (d) $y$-range $1$–$9$, so $1$ unit per square works fine on a standard grid.

Quick Review

Step 1

Identify family

Step 2

Choose $x$-values

Step 3

Build table

Step 4

Choose scale

Step 5

Plot + smooth curve

Reminder

Never straight segments

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