Mathematics • Year 9 • Unit 2 • Lesson 15

Which Family Fits the Story?

Five real-world scenarios. Your job: read the description and the numbers, then decide whether the relationship is best modelled by a parabola, circle, hyperbola, or exponential. Justify with first differences or the equation form.

Apply · Real-World Maths

1. Word problems

For each scenario, identify the curve family, write a sample equation if appropriate, and answer the questions. 3 marks each

1.1, Thrown ball. A ball is thrown up. Its height $h$ (m) above release after $t$ seconds is $h = -5(t - 1)^2 + 6$.

(a) Which family does $h$ vs $t$ belong to? Justify from the equation.
(b) State the vertex and what it means physically.
(c) Which way does it open, and why does that make sense?

Stuck? Squared bracket $\Rightarrow$ parabola. Sign of $a$ gives direction.

1.2, Pizza per person. A pizza is shared between $n$ people, giving $s = 24/n$ slices each.

(a) Which family does $s$ vs $n$ belong to? Justify from the equation.
(b) Build a quick table at $n = 1, 2, 3, 4, 6, 8, 12$.
(c) Which quadrant of the graph is physically meaningful here? (Hint: can $n$ be negative or zero?)

Stuck? $24/n$ form $\Rightarrow$ hyperbola. Only Q1 ($n > 0$) is real-world.

1.3, Drone safe-zone. A drone hovers at the origin and is safe to fly anywhere within $50$ metres horizontally (use a 2-D coordinate grid).

(a) Write the equation describing the edge of the safe zone.
(b) Which family is it?
(c) Is a launch point at $(30, 40)$ inside, on, or outside the safe zone? Show working.

Stuck? Edge of safe zone is a CIRCLE. $r = 50$ so $r^2 = 2500$. Compare $30^2 + 40^2 = 2500$ with $2500$.

1.4, Chain text. One student sends a chain text. Each round, every recipient sends it to TWO new students. So after $r$ rounds, the new-recipients count is $N = 2^r$.

(a) Which family does $N$ vs $r$ belong to? Justify.
(b) Build a table for $r = 0, 1, 2, 3, 4, 5, 6, 7$.
(c) After how many rounds does $N$ first exceed $100$?

Stuck? $2^7 = 128$ is the first power of $2$ that exceeds $100$.

1.5, Mixed table identification. Four classmates collected data and each table is supposedly one of: linear, parabola, hyperbola, or exponential. Identify which is which.

Table A: $x = 1, 2, 3, 4$; $y = 12, 6, 4, 3$.
Table B: $x = 0, 1, 2, 3$; $y = 5, 8, 11, 14$.
Table C: $x = 0, 1, 2, 3$; $y = 1, 4, 9, 16$.
Table D: $x = 0, 1, 2, 3$; $y = 1, 5, 25, 125$.

Stuck? Check first differences. Constant $\Rightarrow$ linear. Squared pattern $\Rightarrow$ parabola. Ratio constant $\Rightarrow$ exponential. $xy = $ constant $\Rightarrow$ hyperbola.

2. Explain your thinking

Use full sentences, no dot points. 4 marks

2.1 A classmate sees both $y = x^2$ and $y = 2^x$ and says "they're basically the same, both have a $2$ and an $x$, just rearranged." In your own words, explain (i) the fundamental difference (where is the variable in each?), (ii) what FAMILY each belongs to, (iii) what is different about their $y$-intercepts, and (iv) compute $y$ for each at $x = 5$ to show how far apart they get.

Stuck? Revisit lesson § "Common Pitfalls", the "Confusing $x^2$ with $2^x$" row says exactly what to ask.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers, Do not peek before attempting

1.1, Thrown ball

(a) Parabola. The squared bracket $(t - 1)^2$ is the signature.
(b) Vertex $(1, 6)$, at $t = 1$ s, the ball reaches its peak height of $6$ m.
(c) Opens DOWN ($a = -5 < 0$). This matches reality: gravity brings the ball back down after the peak.

1.2, Pizza per person

(a) Hyperbola. The $24/n$ form has $n$ in the denominator, the inverse-variation signature.
(b) $n = 1, 2, 3, 4, 6, 8, 12$ gives $s = 24, 12, 8, 6, 4, 3, 2$.
(c) Only Q1 is physically meaningful: $n$ must be a positive whole number (you can't have zero or negative people sharing).

1.3, Drone safe-zone

(a) $\mathbf{x^2 + y^2 = 2500}$.
(b) Circle, centre origin, radius $50$ m.
(c) $30^2 + 40^2 = 900 + 1600 = 2500 = r^2$. So $(30, 40)$ lies exactly ON the boundary of the safe zone.

1.4, Chain text

(a) Exponential. Variable $r$ is in the exponent.
(b) $r = 0, 1, 2, 3, 4, 5, 6, 7$ gives $N = 1, 2, 4, 8, 16, 32, 64, 128$.
(c) $N$ first exceeds $100$ at $r = 7$ rounds ($128 > 100$, while $64 \le 100$ at $r = 6$).

1.5, Mixed table ID

Table A: $xy = 12, 12, 12, 12$ (constant product) $\Rightarrow$ hyperbola ($y = 12/x$).
Table B: first differences $3, 3, 3$ (constant) $\Rightarrow$ linear ($y = 3x + 5$).
Table C: $y$ values $1, 4, 9, 16$ = $1^2, 2^2, 3^2, 4^2$, but careful, $x$ starts at $0$. Check: at $x = 0$, $y = 1$, at $x = 1$, $y = 4$. So $y = (x+1)^2$ would fit. The first differences $3, 5, 7$ are not constant but SECOND differences are constant at $2$, signature of a parabola.
Table D: ratios $5/1, 25/5, 125/25 = 5, 5, 5$ (constant ratio) $\Rightarrow$ exponential ($y = 5^x$).

2.1, Explain your thinking (sample response)

My classmate is wrong because where you put the variable matters enormously. In $y = x^2$, the variable $x$ is the BASE and the $2$ is the exponent, this is a parabola. In $y = 2^x$, the $2$ is the base and the variable $x$ is the EXPONENT, this is an exponential. Their $y$-intercepts already disagree: $y = x^2$ passes through $(0, 0)$ (since $0^2 = 0$), but $y = 2^x$ passes through $(0, 1)$ (since $2^0 = 1$). And the values pull apart fast: at $x = 5$, $y = x^2 = 25$ but $y = 2^x = 32$. By $x = 10$ the parabola gives $100$ but the exponential gives $1024$, ten times bigger. Same digits, totally different families.

Marking: 1 mark for "base vs exponent"; 1 mark for naming parabola vs exponential; 1 mark for the differing $y$-intercepts; 1 mark for the $x = 5$ comparison.