Mathematics • Year 9 • Unit 2 • Lesson 11
Vertex Form in the Real World
Use $y = a(x - h)^2 + k$ to describe a basketball arc, a profit curve, a bridge cable, a roller-coaster dip and a paper aeroplane. Pull the parabola's three numbers out of each story.
1. Word problems
For each scenario, read the vertex-form equation, pull out $a$, $h$ and $k$, and answer in full sentences with units. 3 marks each
1.1 — Basketball arc. The height $h$ (metres) of a basketball $t$ seconds after release is modelled by $h = -5(t - 1)^2 + 6$, valid for $0 \le t \le 2.1$.
(a) State $a$, the vertex, and what the vertex means in plain English.
(b) What is the maximum height of the ball, and at what time does it reach it?
(c) Does the parabola open up or down? Why does that make physical sense here?
1.2 — Daily profit. A school canteen's daily profit $P$ (dollars) when they sell $n$ extra muffins above the usual amount is modelled by $P = -2(n - 20)^2 + 150$.
(a) State the vertex and explain what each coordinate means.
(b) What is the maximum daily profit?
(c) How many extra muffins should they sell to maximise profit?
1.3 — Suspension bridge cable. The shape of a suspension bridge cable above the road can be modelled by $y = 0.02(x - 50)^2 + 4$, where $x$ is the horizontal distance from the left tower (metres) and $y$ is the height above the road (metres).
(a) State the vertex. What does it represent physically?
(b) Is the lowest point of the cable a MIN or a MAX? Justify using the sign of $a$.
(c) How high is the cable above the road at its lowest point?
1.4 — Roller-coaster dip. A roller-coaster track's height $y$ (metres) at horizontal position $x$ (metres) is $y = \tfrac{1}{4}(x - 30)^2 + 5$ for $20 \le x \le 40$.
(a) State the vertex and the axis of symmetry.
(b) Is the lowest point on this stretch the vertex or one of the endpoints?
(c) Find the height of the track at $x = 30$ and at $x = 40$. Which is higher?
1.5 — Paper aeroplane drop. A paper aeroplane is thrown from a balcony. Its height $h$ (metres) after $t$ seconds is modelled by $h = -1(t - 2)^2 + 8$.
(a) State the vertex. When does the plane reach its highest point and what is that height?
(b) Expand the equation into the form $h = at^2 + bt + c$.
(c) Use the expanded form to find the initial height $h$ when $t = 0$.
2. Explain your thinking
Use full sentences, no dot points. 4 marks
2.1 A classmate looks at the equation $y = 2(x + 3)^2 - 5$ and says "the vertex is at $(3, -5)$ because the numbers are $3$ and $-5$." In your own words, explain (i) why their answer is wrong, (ii) what the correct vertex actually is, (iii) the general rule for reading $h$ from a vertex-form bracket, and (iv) give one similar example (you choose the equation) where the sign of $h$ might also trick someone.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Basketball arc
(a) $a = -5$ (opens DOWN), vertex $(1, 6)$. The vertex means: at $t = 1$ second after release, the ball reaches its highest point of $6$ m.
(b) Maximum height $= 6$ m, reached at $t = 1$ s.
(c) Opens DOWN. Physically, a thrown ball goes up, then comes back down under gravity — the parabola shape opens downward with the peak at the top.
1.2 — Daily profit
(a) Vertex $(20, 150)$. $n = 20$: the optimum number of extra muffins to sell. $P = 150$: the maximum daily profit (in dollars).
(b) Maximum profit $= \$150$ per day.
(c) Sell 20 extra muffins beyond the usual amount.
1.3 — Suspension bridge cable
(a) Vertex $(50, 4)$. Physically: the lowest point of the cable is $50$ m along (halfway between the towers, presumably) and sits $4$ m above the road.
(b) MIN. $a = 0.02 > 0$ means the parabola opens UP, so the vertex is at the bottom — the lowest point.
(c) $4$ m above the road.
1.4 — Roller-coaster dip
(a) Vertex $(30, 5)$, axis $x = 30$.
(b) The vertex is the lowest point (since $a = \tfrac{1}{4} > 0$, the parabola opens UP).
(c) At $x = 30$: $y = \tfrac{1}{4}(0)^2 + 5 = 5$ m. At $x = 40$: $y = \tfrac{1}{4}(10)^2 + 5 = \tfrac{1}{4}(100) + 5 = 25 + 5 = 30$ m. The point at $x = 40$ is far higher.
1.5 — Paper aeroplane drop
(a) Vertex $(2, 8)$. The plane peaks at height $8$ m at $t = 2$ s.
(b) $(t - 2)^2 = t^2 - 4t + 4$. Multiply by $-1$: $-t^2 + 4t - 4$. Add $8$: $\mathbf{h = -t^2 + 4t + 4}$.
(c) At $t = 0$: $h = -0 + 0 + 4 = \mathbf{4}$ m initial height.
2.1 — Explain your thinking (sample response)
My classmate is wrong because they have forgotten the sign flip on $h$. Vertex form is $y = a(x - h)^2 + k$. In the given equation we have $(x + 3)$, which we have to rewrite as $(x - (-3))$ before we can read $h$. That means $h = -3$, not $3$. The constant $-5$ on the end is $k = -5$ — that part the classmate got right. So the correct vertex is $\mathbf{(-3, -5)}$. The general rule is: the sign in front of $h$ FLIPS when you read it from inside the bracket. Plus inside means negative $h$; minus inside means positive $h$. A similar tricky example is $y = (x + 7)^2 + 1$ — many students would write the vertex as $(7, 1)$, but it is actually $(-7, 1)$.
Marking: 1 mark for naming the flip-sign rule; 1 mark for the correct vertex $(-3, -5)$; 1 mark for clear general explanation; 1 mark for a valid similar example.