Mathematics • Year 9 • Unit 2 • Lesson 3
Stretching and Squashing Parabolas
Use $y = ax^2$ to model satellite dishes, soccer kicks, fountain jets, designer skate ramps and headlight reflectors. Then explain in your own words why a fraction $a$ makes the curve wider, not narrower.
1. Word problems
Each problem uses $y = ax^2$ to describe something physical. Show your working, including the "square first, then multiply" order. 3 marks each
1.1, Satellite dish profile. A small satellite dish has cross-section $y = \tfrac{1}{4}x^2$ (where $x$ is horizontal distance in metres from the centre, and $y$ is the dish depth in metres). It needs to reach a depth of $y = 1$ m at its edge.
(a) Find the value(s) of $x$ where $y = 1$.
(b) State the total width of the dish (edge to edge).
(c) Is this dish wider or narrower than the reference curve $y = x^2$? One-sentence reason.
1.2, Soccer ball table. A child kicks a soccer ball straight up. Its height above the start point (in metres) at time $t$ (in seconds before the ball reaches its peak) is modelled by $h = 5t^2$ (taking $t = 0$ as the peak, then $t$ counts down to the moment of release).
(a) Build a table for $t = 0, 1, 2$.
(b) Compared to the reference $y = x^2$, is the soccer-ball curve wider or narrower? Why?
(c) Find $t$ when $h = 20$ metres.
1.3, Fountain water arch. A garden fountain shoots water from a central nozzle. The water's path on each side is modelled by $y = -\tfrac{1}{8}x^2 + 2$ for the upward arc, but a designer is approximating it just with the dilation part: $y = \tfrac{1}{8}x^2$ (for the falling cone shape).
(a) For the dilation $y = \tfrac{1}{8}x^2$, find $y$ when $x = 4$.
(b) Find $y$ when $x = 8$, how many times bigger than your $x = 4$ answer is it?
(c) Is $y = \tfrac{1}{8}x^2$ wider or narrower than $y = x^2$?
1.4, Designer skate ramp. A skate ramp's cross-section is modelled by $y = ax^2$. The designer wants the ramp to be $1$ metre high when $x = 2$ metres from the centre line.
(a) Find $a$ so that the ramp passes through $(2, 1)$.
(b) Write the ramp's equation.
(c) Compared to a standard $y = x^2$ ramp, is this one wider or narrower? Why does that make it safer for beginners?
1.5, Headlight reflector. A car's parabolic headlight reflector has cross-section $y = 3x^2$ (with $x$ in cm). Compare it with a wider truck headlight $y = \tfrac{1}{2}x^2$ at the same horizontal distance $x = 2$ cm.
(a) Compute $y$ for both reflectors at $x = 2$ cm.
(b) Which reflector is deeper at that horizontal distance, and by how many cm?
(c) Which reflector's curve is narrower, the car or the truck, and what does that mean for how tightly the headlight beam focuses?
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate says "$y = \tfrac{1}{2}x^2$ must be narrower than $y = x^2$ because $\tfrac{1}{2}$ is a small number, and small things are skinny." In your own words, explain (i) what the classmate has confused, (ii) what $a = \tfrac{1}{2}$ actually does to every $y$-value compared with $y = x^2$, and (iii) why halving the $y$-values produces a WIDER curve, not a narrower one.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Satellite dish
(a) Set $y = 1$: $1 = \tfrac{1}{4}x^2 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$. So the edges are at $x = -2$ m and $x = 2$ m.
(b) Total width: from $-2$ to $+2$ is $\mathbf{4}$ m.
(c) $a = \tfrac{1}{4} < 1$, so this dish is wider than the reference $y = x^2$, the curve spreads out more for the same depth.
1.2, Soccer ball table
(a) $t = 0$: $h = 0$. $t = 1$: $h = 5$. $t = 2$: $h = 20$.
(b) $a = 5 > 1$, so narrower than $y = x^2$. The ball gains height fast as $t$ increases (because gravity is fairly strong in the model).
(c) $20 = 5t^2 \Rightarrow t^2 = 4 \Rightarrow t = 2$ s (positive root, since time can't be negative).
1.3, Fountain water arch
(a) $y = \tfrac{1}{8}(4)^2 = \tfrac{16}{8} = \mathbf{2}$ m.
(b) $y = \tfrac{1}{8}(8)^2 = \tfrac{64}{8} = \mathbf{8}$ m. That's $\dfrac{8}{2} = 4$ times bigger, matches our prediction that doubling $x$ quadruples $y$.
(c) $a = \tfrac{1}{8} < 1$, so the curve is wider than $y = x^2$.
1.4, Designer skate ramp
(a) Sub $(2, 1)$: $1 = a(2)^2 = 4a$, so $a = \mathbf{\tfrac{1}{4}}$.
(b) Equation: $y = \tfrac{1}{4}x^2$.
(c) $a = \tfrac{1}{4} < 1$, so the ramp curve is wider than $y = x^2$, the rise is more gradual for any given horizontal distance, meaning the ramp is less steep, which is safer for beginners learning balance.
1.5, Headlight reflector
(a) Car: $y = 3(2)^2 = 12$ cm. Truck: $y = \tfrac{1}{2}(2)^2 = 2$ cm.
(b) The car reflector is deeper, by $12 - 2 = \mathbf{10}$ cm at $x = 2$ cm.
(c) The car reflector is narrower ($a = 3 > 1$). A narrower parabola focuses incoming light to a tighter point (closer to the focus), giving a more concentrated beam, useful for a car headlight aimed straight down the road.
2.1, Explain your thinking (sample response)
My classmate has confused the size of the coefficient $a$ with the appearance of the curve, "small number" does not mean "skinny curve". In $y = ax^2$ the value of $a$ is a multiplier applied to every $y$-value. When $a = \tfrac{1}{2}$, every $y$-value of $y = x^2$ is halved, for example at $x = 2$, $y = x^2$ gives $y = 4$, but $y = \tfrac{1}{2}x^2$ gives only $y = 2$. Halving the heights flattens the curve closer to the $x$-axis, which makes its arms spread out further from the $y$-axis before they reach any given height, producing a wider parabola, not a narrower one. The rule from Lesson 3 is: bigger $a$ means narrower; smaller positive $a$ (between $0$ and $1$) means wider.
Marking: 1 mark for naming the confusion; 1 mark for "$a$ multiplies every $y$-value"; 1 mark for the wider-not-narrower conclusion; 1 mark for clarity and a worked example.