Mathematics • Year 9 • Unit 2 • Lesson 13
Hyperbolas — Mixed Challenge
Pull together every hyperbola skill: read values, name asymptotes, compare two hyperbolas, find $k$ from a point, and solve a hyperbola-meets-line system. Then catch a quadrant-sign mistake and invent your own inverse-variation table.
1. Mixed problems
Show working. 3 marks each
1.1 For $y = 12/x$, complete the table at $x = -6, -3, -2, -1, 1, 2, 3, 6$. State the asymptotes and the two quadrants of the branches.
1.2 A hyperbola $y = k/x$ passes through $(3, 5)$. (a) Find $k$. (b) Write the equation. (c) Find $y$ when $x = 15$.
1.3 Compare the two hyperbolas $y = 4/x$ and $y = 16/x$. (a) Do they share asymptotes? Justify. (b) At $x = 2$, which gives a $y$ further from the $x$-axis? (c) In general, how does increasing $|k|$ change the position of the branches relative to the origin?
1.4 Classify each into the correct quadrants of branches: (a) $y = 9/x$ (b) $y = -9/x$ (c) $y = 1/x$ (d) $y = -100/x$. State Q1+Q3 or Q2+Q4 for each.
1.5 The hyperbola $y = 6/x$ and the line $y = x + 1$ meet at one or more points. Use substitution to find the $x$-values where they meet (you should get a quadratic). Then state the meeting points.
1.6 Sketch (roughly, in the margin or your book) the curves $y = 4/x$ and $y = -4/x$ on the same axes. Describe what transformation takes one curve to the other and state the four quadrants between them (which of Q1, Q2, Q3, Q4 contain a branch of EITHER hyperbola).
2. Find the mistake
Another student has classified four hyperbolas into their quadrants. Their working is shown. Two of the four are wrong. Find them, explain, and fix them. 3 marks
Student's classifications:
A: $y = 8/x$ → branches in Q1 and Q3 ✓
B: $y = -2/x$ → branches in Q1 and Q3 (because $-2/-1 = 2 > 0$, so Q1)
C: $y = -5/x$ → branches in Q2 and Q4 ✓
D: $y = 3/x$ → branches in Q2 and Q4 (because positive $k$ goes in even quadrants)
(a) Which two classifications are wrong?
(b) For each wrong one, explain in one sentence what the student is muddling up.
(c) Write out the corrected quadrant pair for each wrong one.
Stuck? The rule is simple: $k > 0 \Rightarrow$ Q1 and Q3 (same-sign coordinates); $k < 0 \Rightarrow$ Q2 and Q4 (opposite-sign coordinates).3. Open-ended challenge — invent an inverse-variation story
This question has many valid answers. Be creative. 4 marks
3.1 Invent a real-world story (DIFFERENT from the worksheet examples: pizza, speed, brightness, gears, pressure) where two quantities $x$ and $y$ are connected by inverse variation $y = k/x$.
For your story:
(i) Describe the situation in 1–2 sentences and clearly name what $x$ and $y$ are (including their units).
(ii) State the value of $k$ you have chosen, and what $k$ means physically in your scenario.
(iii) Build a table of at least FOUR pairs $(x, y)$ from your equation.
(iv) Predict what happens to $y$ if $x$ doubles and what happens if $x$ halves — using one specific pair from your table to demonstrate.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Table for $y = 12/x$
$x = -6, -3, -2, -1, 1, 2, 3, 6$ gives $y = -2, -4, -6, -12, 12, 6, 4, 2$.
Asymptotes: $x = 0$ and $y = 0$. Branches in Q1 and Q3 ($k = 12 > 0$).
1.2 — $y = k/x$ through $(3, 5)$
(a) $5 = k/3 \Rightarrow k = 15$.
(b) Equation: $\mathbf{y = 15/x}$.
(c) At $x = 15$: $y = 15/15 = 1$.
1.3 — $y = 4/x$ vs $y = 16/x$
(a) Yes, both have asymptotes $x = 0$ and $y = 0$ (true of every $y = k/x$).
(b) At $x = 2$: $y = 4/2 = 2$ vs $y = 16/2 = 8$. The $y = 16/x$ curve is further from the $x$-axis there (8 vs 2).
(c) Increasing $|k|$ pushes the branches further from the origin — they "bulge out" more. Smaller $|k|$ keeps the branches closer in toward the axes.
1.4 — Classify quadrants
(a) $k = 9 > 0$: Q1 and Q3.
(b) $k = -9 < 0$: Q2 and Q4.
(c) $k = 1 > 0$: Q1 and Q3.
(d) $k = -100 < 0$: Q2 and Q4.
1.5 — Hyperbola meets line
$x + 1 = 6/x \Rightarrow x(x + 1) = 6 \Rightarrow x^2 + x - 6 = 0 \Rightarrow (x + 3)(x - 2) = 0$.
So $x = -3$ or $x = 2$. At $x = -3$: $y = -3 + 1 = -2$, point $(-3, -2)$. At $x = 2$: $y = 2 + 1 = 3$, point $(2, 3)$.
Check both on $y = 6/x$: $6/(-3) = -2$ ✓ and $6/2 = 3$ ✓. Meeting points: $\mathbf{(-3, -2)}$ and $\mathbf{(2, 3)}$.
1.6 — $y = 4/x$ vs $y = -4/x$
$y = 4/x$ has branches in Q1 and Q3. $y = -4/x$ has branches in Q2 and Q4. The transformation taking one to the other is a reflection in the $x$-axis (or equivalently, a reflection in the $y$-axis — both produce the same flip for $y = k/x$). Between them, all four quadrants Q1, Q2, Q3, Q4 contain a branch of one of the two hyperbolas.
2 — Find the mistake
(a) Wrong ones are B and D.
(b) B: the student checked just one point ($x = -1$) and saw $y = -2/(-1) = 2$ — but that point $(-1, 2)$ is in Q2, not Q1. They mixed up "positive $y$" with "Q1". D: the student invented a "positive $k$ goes in even quadrants" rule — that's not the lesson rule. The actual rule is positive $k$ $\Rightarrow$ Q1 and Q3 (same-sign coordinates).
(c) Corrected: B: $y = -2/x$ has $k = -2 < 0$, so branches in Q2 and Q4. D: $y = 3/x$ has $k = 3 > 0$, so branches in Q1 and Q3.
3 — Open-ended challenge (sample solution)
Story: A class of $n$ students is given a 24-cookie tray to share equally. Each student gets $c = 24/n$ cookies.
(i) $n$ = number of students (people), $c$ = cookies per student (cookies).
(ii) $k = 24$ = total cookies on the tray.
(iii) Table: $n = 1, 2, 3, 4, 6, 8$ gives $c = 24, 12, 8, 6, 4, 3$.
(iv) From the pair $(4, 6)$ — doubling $n$ from $4$ to $8$ halves $c$ from $6$ to $3$. Halving $n$ from $4$ to $2$ doubles $c$ from $6$ to $12$.
Marking: 1 mark for a distinct story not from the worksheet examples; 1 mark for clear variables + units; 1 mark for valid table from chosen $k$; 1 mark for correct doubling/halving prediction.