Lesson 13 ~25 min Unit 2 · Non-Linear +85 XP

The Hyperbola $y = \dfrac{k}{x}$

An inverse relationship with two curving branches that NEVER touch the axes. Meet asymptotes and learn to spot which way the branches bend.

Today's hook: Try $x = 0$ in $y = 6/x$. What happens? Why doesn't the graph cross the $y$-axis?

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Think First

warm-up

In $y = k/x$, $y$ and $x$ are INVERSELY related: as one grows, the other shrinks. Take $y = 6/x$. Make a quick table for $x = 1, 2, 3, 6$ and for $x = -1, -2, -3, -6$. What happens for $x = 0$? Why?

Record your answer in your workbook.

The Big Idea

+5 XP

The hyperbola $y = k/x$ is the shape of an INVERSE relationship: $xy = k$. It has TWO branches, both bending towards the axes but never touching them. Those axes are ASYMPTOTES.

The red curve $y = 6/x$ has $k = 6 > 0$, so its two branches sit in quadrants 1 and 3 (both $x, y$ positive, or both negative). Both branches hug closer and closer to the $x$-axis as $x \to \pm \infty$, and closer to the $y$-axis as $x \to 0$, but they never cross either axis.

$y = \dfrac{k}{x}$, asymptotes $x = 0$ and $y = 0$.

$xy = k$

Product is constant, the inverse relation.

Two branches

$k > 0$: Q1 + Q3. $k < 0$: Q2 + Q4.

Never touches axes

$x \neq 0$ and $y \neq 0$, asymptotes.

What You'll Master

objectives

Know

$y = k/x$ describes inverse variation: $xy = k$
The graph has TWO branches with asymptotes $x = 0$ and $y = 0$
$k > 0$ gives Q1 + Q3; $k < 0$ gives Q2 + Q4

Understand

Why $x = 0$ is forbidden (division by zero is undefined)
Why branches never CROSS the axes, they approach but don't touch
Why larger $|k|$ pushes branches further from the origin

Can Do

Build a table of values and plot a hyperbola
State asymptotes and which quadrants the branches sit in
Decide whether a given point lies on $y = k/x$

Words You Need

vocabulary

HyperbolaThe curve $y = k/x$ (rectangular hyperbola with asymptotes on the axes).

Inverse variationTwo variables related by $xy = $ constant. Doubling $x$ halves $y$.

AsymptoteA line a curve gets arbitrarily close to but never touches.

BranchOne of the two continuous arcs that make up a hyperbola.

Undefined$y = k/x$ at $x = 0$: division by zero has no value, so the graph has a gap there.

QuadrantOne of the four regions of the coordinate plane: Q1 ($+$, $+$), Q2 ($-$, $+$), Q3 ($-$, $-$), Q4 ($+$, $-$).

Spot the Trap

heads-up

Wrong: Drawing $y = 6/x$ as a single connected curve crossing the axes.

Right: TWO separate branches. Neither touches an axis. There's a gap at $x = 0$.

Wrong: Putting both branches of $y = -4/x$ in quadrants 1 and 3.

Right: $k = -4 < 0$, so the branches sit in Q2 (top-left) and Q4 (bottom-right). $x$ and $y$ have OPPOSITE signs.

Plotting from a Table

+5 XP

The cleanest way to sketch $y = k/x$ is a table of values using divisors of $k$:

For $y = 6/x$: take $x = \pm 1, \pm 2, \pm 3, \pm 6$. Then $y$ values come out as integers: $6, 3, 2, 1, -6, -3, -2, -1$. Plot 8 points, join with TWO smooth curves (one per side), watching them approach the axes but never touch.

Don't pick $x = 0$, it's undefined. Note the symmetry: if $(a, b)$ is on the curve, so is $(-a, -b)$.

Table $\to$ pick easy $x$ values $\to$ two smooth branches.

Use divisors of $k$

Clean integer points: e.g. $\pm 1, \pm 2, \pm 3, \pm 6$ for $k = 6$.

Both signs of $x$

Need positives AND negatives to see both branches.

Symmetry about origin

$(a, b)$ and $(-a, -b)$ both on the curve.

Effect of $k$, Sign and Size

+5 XP

Two questions about $k$:

Sign of $k$: $k > 0$ $\to$ branches in Q1 + Q3 (both coordinates same sign). $k < 0$ $\to$ branches in Q2 + Q4 (opposite signs).
Size of $|k|$: larger $|k|$ pushes the branches FURTHER from the origin. e.g. $y = 1/x$ passes through $(1, 1)$; $y = 8/x$ passes through $(1, 8)$, further out.

Asymptotes ($x = 0$ and $y = 0$) are the SAME for every $y = k/x$, regardless of $k$.

Sign of $k$ $\to$ quadrants. Size of $|k|$ $\to$ distance from origin.

Same asymptotes

Always $x = 0$ and $y = 0$ for $y = k/x$.

Sign flips quadrants

$k$ flipping sign reflects the curve across an axis.

$y = 1/x$ basic

Passes through $(1, 1)$ and $(-1, -1)$.

Watch Me Solve It · 3 examples

Watch Me Solve It · Table and sketch for $y = 6/x$

+15 XP per step

PROBLEM

Build a table for $y = 6/x$ at $x = \pm 1, \pm 2, \pm 3, \pm 6$, then describe the sketch (branches, asymptotes).

1
Positive $x$ side
$x = 1, y = 6$. $x = 2, y = 3$. $x = 3, y = 2$. $x = 6, y = 1$.
2
Negative $x$ side
$x = -1, y = -6$. $x = -2, y = -3$. $x = -3, y = -2$. $x = -6, y = -1$.
3
Describe sketch
$k = 6 > 0$: branches in Q1 and Q3. Both branches approach the axes (asymptotes $x = 0, y = 0$) but never touch.
Symmetric: $(a, b)$ and $(-a, -b)$ are mirror images through the origin.

AnswerBranches in Q1 and Q3, asymptotes $x = 0$ and $y = 0$.

Watch Me Solve It · Effect of negative $k$

+15 XP per step

PROBLEM

For $y = -4/x$, state the quadrants of the branches, the asymptotes, and one point on each branch.

1
Sign of $k$
$k = -4 < 0$, so branches are in Q2 (top-left) and Q4 (bottom-right).
2
Asymptotes
$x = 0$ and $y = 0$ (same as always for $y = k/x$).
3
Sample points
$x = 1: y = -4$ (Q4). $x = -1: y = 4$ (Q2).
Opposite-sign coordinates, the trademark of $k < 0$.

AnswerBranches in Q2 and Q4. Asymptotes $x = 0, y = 0$. Points: $(1, -4)$, $(-1, 4)$.

Watch Me Solve It · Find $k$ from a point

+15 XP per step

PROBLEM

A hyperbola $y = k/x$ passes through $(2, 5)$. Find $k$ and state the equation.

1
Substitute the point
$y = k/x \Rightarrow 5 = k/2$.
2
Solve for $k$
Multiply both sides by 2: $k = 10$.
3
Write the equation
$y = 10/x$.
Quick check: at $x = 2$, $y = 10/2 = 5$. Matches.

Answer$k = 10$; equation $y = 10/x$.

Common Pitfalls

heads-up

Crossing the axes

Drawing the hyperbola as a single curve that passes through the origin.

Fix: TWO separate branches. Neither touches an axis. There is a gap at $x = 0$.

Wrong quadrants for $k < 0$

Putting $y = -2/x$ in Q1 and Q3.

Fix: $k < 0$ puts branches in Q2 and Q4 (opposite-sign coordinates).

Including $x = 0$ in the table

Writing "$x = 0, y = 0$" or "$y = $ infinity".

Fix: $y$ at $x = 0$ is UNDEFINED. Skip it.

Copy Into Your Books

Equation

$y = k/x$
$xy = k$ (inverse)
$x \neq 0$

Asymptotes

$x = 0$ (y-axis)
$y = 0$ (x-axis)
Never touched

Branches

$k > 0$: Q1, Q3
$k < 0$: Q2, Q4
Symmetric thru origin

Table tips

Use divisors of $k$
Both signs of $x$
Skip $x = 0$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Hyperbola Drills

4 problems

Four quick hyperbola questions.

1 For $y = 12/x$, find $y$ when $x = 4$.
$y = 12/4$.$y = 3$
2 State the asymptotes of $y = -8/x$.
Same for every $y = k/x$.$x = 0$ and $y = 0$
3 Which quadrants contain the branches of $y = -10/x$?
$k = -10 < 0$.Q2 and Q4
4 A hyperbola $y = k/x$ passes through $(3, 4)$. Find $k$.
$4 = k/3 \Rightarrow k = 12$.$y = 12/x$

Complete in your workbook.

Quick Check · 5 questions

The asymptotes of $y = 6/x$ are:

+10 XP

The branches of $y = -4/x$ sit in:

+10 XP

If $y = 8/x$, then at $x = 2$, $y = $:

+10 XP

For $y = k/x$ at $x = 0$, the value of $y$ is:

+10 XP

A hyperbola $y = k/x$ passes through $(3, 5)$. The value of $k$ is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy3 MARKS

Q6. Complete the table for $y = 12/x$ using $x = -6, -3, -2, -1, 1, 2, 3, 6$. State which quadrants contain branches and the asymptotes.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. A hyperbola $y = k/x$ passes through $(-2, 6)$. (a) Find $k$. (b) State the equation. (c) Which quadrants do the branches sit in?

Answer in your workbook.

ReasonHard3 MARKS

Q8. Two hyperbolas: $y = 4/x$ and $y = 16/x$. (a) Do they share asymptotes? Justify. (b) Compare the points at $x = 2$ for each, which is further from the $x$-axis? (c) Hence describe how increasing $|k|$ changes the position of the branches.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B asymptotes are $x = 0$ and $y = 0$.

2. D$k < 0$: branches in Q2 and Q4.

3. A$y = 8/2 = 4$.

4. C undefined (division by zero).

5. B$5 = k/3 \Rightarrow k = 15$.

Show Your Working Model Answers

Q6 (3 marks): Table: $(-6, -2), (-3, -4), (-2, -6), (-1, -12), (1, 12), (2, 6), (3, 4), (6, 2)$ [1]. $k = 12 > 0$ so branches in Q1 and Q3 [1]. Asymptotes $x = 0$ and $y = 0$ [1].

Q7 (3 marks): (a) $6 = k/(-2) \Rightarrow k = -12$ [1]. (b) $y = -12/x$ [1]. (c) $k < 0$, so branches in Q2 and Q4 [1].

Q8 (3 marks): (a) Both have asymptotes $x = 0, y = 0$ (true for every $y = k/x$) [1]. (b) At $x = 2$: $y = 4/2 = 2$ vs $y = 16/2 = 8$. So $y = 16/x$ is further from the $x$-axis [1]. (c) Larger $|k|$ pushes the branches further from the origin / from the asymptotes [1].

Stretch Challenge · +25 XP, +10 coins

Where Hyperbola Meets Line

Find any intersection points of $y = 6/x$ and $y = x$. (a) Set them equal: $6/x = x$. (b) Multiply both sides by $x$ (noting $x \neq 0$): $6 = x^2$. (c) Solve: $x = \pm\sqrt{6}$. (d) Find matching $y$ values. (e) Explain what symmetry of the answers tells you about the geometry of the two curves.

Reveal solution

(a) $6/x = x$. (b) $x^2 = 6$ (with $x \neq 0$). (c) $x = \pm\sqrt{6}$. (d) Matching $y = x$: $y = \pm\sqrt{6}$. Points $(\sqrt{6}, \sqrt{6})$ and $(-\sqrt{6}, -\sqrt{6})$. (e) Both points are symmetric about the origin, consistent with both $y = x$ (line through origin) and $y = 6/x$ (hyperbola symmetric about origin) sharing that symmetry.

Quick Review

Equation

$y = k/x$

Asymptotes

$x = 0$, $y = 0$

$k > 0$

Q1 and Q3

$k < 0$

Q2 and Q4

$x = 0$

Undefined

Symmetry

$(a, b) \leftrightarrow (-a, -b)$

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