Mathematics • Year 9 • Unit 2 • Lesson 15
Comparing Non-Linear Graphs — Mixed Challenge
Pull together the entire unit: sort equations into the four families, fill a comparison table of key features, sketch and overlay two curves, find a meeting point, catch a misclassification, and design your own four-family quiz.
1. Mixed problems
Show working. 3 marks each
1.1 Sort each into one of the four families (parabola, circle, hyperbola, exponential): (a) $y = -2(x - 3)^2 + 1$ (b) $y = -5/x$ (c) $x^2 + y^2 = 1$ (d) $y = 4^x$ (e) $y = x^2 + x + 1$ (f) $y = (\tfrac{1}{3})^x$.
1.2 Complete a comparison table for $y = 4/x$, $y = x^2$ and $x^2 + y^2 = 4$. State (a) the $y$-intercept(s) of each, (b) the $x$-intercept(s) of each (or "none"), (c) which has asymptotes and what those asymptotes are.
1.3 Sketch $y = x^2$ and $y = 2^x$ on the SAME axes for $-2 \le x \le 3$ (rough sketch in the margin is fine). Label the $y$-intercept of each, and identify the two integer points where they share a $y$-value.
1.4 A relationship gives $x: 0, 1, 2, 3, 4$ and $y: -1, 2, 5, 8, 11$. (a) Show using first differences that this is linear. (b) Write a rule of the form $y = mx + c$. (c) Why is this NOT in any of the four non-linear families?
1.5 The hyperbola $y = 4/x$ and the line $y = x$ meet where? Use substitution to find the meeting $x$-values and the corresponding meeting points.
1.6 For each family, give one example of an equation and ONE distinguishing feature that NO other family in this unit has: (a) parabola, (b) circle, (c) hyperbola, (d) exponential.
2. Find the mistake
A student has tried to classify five equations into families. Their work is below. Two classifications are wrong. Find them, explain, and correct. 3 marks
Student's classifications:
A: $y = 2x + 5$ → Linear ✓
B: $x^2 + y^2 = 16$ → Parabola (because it has $x^2$)
C: $y = 2^x$ → Exponential ✓
D: $y = 5/x$ → Hyperbola ✓
E: $y = -x^2 + 4$ → Exponential (because there's a negative power)
(a) Which two classifications are wrong?
(b) For each wrong one, explain in one sentence why the reasoning is mistaken.
(c) Give the corrected family for each wrong one (with one distinguishing feature).
Stuck? B has BOTH $x^2$ and $y^2$ summed — that's the circle signature, not parabola. E has a squared $x$ (variable in the base, not the exponent) — that's parabola, not exponential.3. Open-ended challenge — design a four-family quiz
This question has many valid answers. Be creative. 4 marks
3.1 Design a 4-question identify-the-family quiz for a classmate. Each question must be ONE of: parabola, circle, hyperbola, or exponential (one from each family). Your job:
(i) Write FOUR equations, one from each family. Mix it up — don't use the simplest possible form (so not just $y = x^2$, $x^2 + y^2 = 1$, $y = 1/x$, $y = 2^x$). Choose your own parameters.
(ii) For each, state the family AND give one key feature (vertex, radius, asymptotes/quadrants, or $y$-intercept respectively).
(iii) Write a one-sentence "trap" for each — a wrong family a classmate might pick — and explain how to avoid it.
(iv) Compute $y$ at $x = 2$ for the parabola, hyperbola, and exponential (the circle won't give a single $y$, so skip it for the circle).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sort into families
(a) Parabola (squared bracket).
(b) Hyperbola ($x$ in denominator).
(c) Circle ($x^2 + y^2 = r^2$ form).
(d) Exponential ($x$ in the exponent).
(e) Parabola (still has $x^2$, just in expanded form).
(f) Exponential (variable in the exponent, base $\tfrac{1}{3}$ — this is exponential decay).
1.2 — Comparison table
$y = 4/x$ (hyperbola): (a) $y$-int: none (undefined at $x = 0$). (b) $x$-int: none. (c) Asymptotes $x = 0$ and $y = 0$.
$y = x^2$ (parabola): (a) $y$-int: $(0, 0)$. (b) $x$-int: $(0, 0)$. (c) No asymptotes.
$x^2 + y^2 = 4$ (circle, $r = 2$): (a) $y$-int(s): $(0, 2)$ and $(0, -2)$. (b) $x$-int(s): $(2, 0)$ and $(-2, 0)$. (c) No asymptotes.
1.3 — Sketch $y = x^2$ and $y = 2^x$
Quick values $-2 \le x \le 3$:
$y = x^2$: $4, 1, 0, 1, 4, 9$.
$y = 2^x$: $1/4, 1/2, 1, 2, 4, 8$.
$y$-intercepts: parabola through $(0, 0)$; exponential through $(0, 1)$.
Both curves share $y = 4$ at $x = 2$ (parabola) and $x = 2$ (exponential); both share $y = 1$ at $x = 0$ (exponential) and $x = -1, 1$ (parabola). The two integer matches where they share the same $y$-value are at $(2, 4)$ for both and at integer $y = 1$ where parabola gives $x = 1$, exponential gives $x = 0$ — but the actual integer crossing points (where both curves give the same $y$ at the same $x$) are at $x = 2$ ($y = 4$) and $x = 4$ ($y = 16$ for both). Accept either of these two crossings as the answer.
1.4 — Linear identification
(a) First differences: $3, 3, 3, 3$ — constant $\Rightarrow$ linear.
(b) $m = 3$ (the constant difference); at $x = 0$, $y = -1$, so $c = -1$. Rule: $\mathbf{y = 3x - 1}$.
(c) Linear relationships have $x$ to the power $1$ only — no $x^2$, no $1/x$, no $a^x$. So they don't fit any of the four non-linear families.
1.5 — Hyperbola meets line
$x = 4/x \Rightarrow x^2 = 4 \Rightarrow x = 2$ or $x = -2$. At $x = 2$: $y = 2$, point $(2, 2)$. At $x = -2$: $y = -2$, point $(-2, -2)$. Meeting points: $\mathbf{(2, 2)}$ and $\mathbf{(-2, -2)}$.
1.6 — Distinguishing features
(a) Parabola, e.g. $y = (x - 1)^2 + 2$ — unique feature: a vertex (lowest or highest point) with axis of symmetry.
(b) Circle, e.g. $x^2 + y^2 = 9$ — unique feature: a closed loop with finite size, set by the radius.
(c) Hyperbola, e.g. $y = 6/x$ — unique feature: TWO disconnected branches with the coordinate axes as asymptotes.
(d) Exponential, e.g. $y = 3^x$ — unique feature: passes through $(0, 1)$ with a horizontal asymptote $y = 0$ and never crosses the $x$-axis. (Hyperbolas also never cross the $x$-axis, but they have two branches and a vertical asymptote too — the exponential has only the horizontal asymptote.)
2 — Find the mistake
(a) Wrong: B and E.
(b) B: the student called $x^2 + y^2 = 16$ a parabola because they spotted $x^2$ — but the equation ALSO contains $y^2$ summed to it, which is the circle signature (not parabola). E: the student called $y = -x^2 + 4$ exponential because "there's a power" — but the power is on the $x$ (variable in the base), which is the parabola signature. Exponential needs the variable in the EXPONENT.
(c) B: Circle, centre $(0, 0)$, radius $4$. E: Parabola (opens down, since coefficient of $x^2$ is negative), vertex at $(0, 4)$.
3 — Open-ended challenge (sample solution)
Quiz set:
Q1: $y = 3(x - 2)^2 + 1$ — parabola; vertex $(2, 1)$. Trap: someone might call it linear because they see "$x$"; avoid by spotting the squared bracket.
Q2: $x^2 + y^2 = 36$ — circle; radius $6$. Trap: someone might call it a parabola for the $x^2$; avoid by noticing BOTH $x^2$ AND $y^2$ summed.
Q3: $y = -12/x$ — hyperbola; branches in Q2 and Q4 ($k = -12 < 0$). Trap: someone might call it linear because $y$ is "$x$ over something"; avoid by spotting the $x$ in the denominator.
Q4: $y = 4^x$ — exponential; $y$-intercept $(0, 1)$. Trap: someone might call it parabolic; avoid by checking where the $x$ sits — the $x$ is the exponent, not the base.
Values at $x = 2$: parabola $3(0)^2 + 1 = 1$; hyperbola $-12/2 = -6$; exponential $4^2 = 16$.
Marking: 1 mark per family with valid equation and key feature, plus 1 mark for sensible traps and $x = 2$ values.