Mathematics • Year 9 • Unit 2 • Lesson 8
Vertex Form in Real Settings
Use vertex form $y = a(x - h)^2 + k$ for genuinely 2D situations: a basketball shot, a parabolic mirror, a profit curve, a roller-coaster loop, and a swimming-pool dive arc.
1. Word problems
For each scenario: identify $a$, $h$, $k$ and answer the questions. Show working. 3 marks each
1.1 — Basketball shot. A basketball follows the path $y = -\tfrac{1}{2}(x - 4)^2 + 8$, where $x$ is the horizontal distance from the shooter (m) and $y$ is the height of the ball above the floor (m).
(a) State the vertex of the path. What does it mean physically?
(b) Does the path have a max or min at the vertex? Why does that match what a basketball does?
(c) How high is the ball when it leaves the shooter's hand at $x = 0$?
1.2 — Parabolic mirror. The cross-section of a parabolic torch reflector is $y = 2(x - 1)^2 - 3$, where $x$ is the horizontal distance (cm) along the base and $y$ is the depth (cm) below the rim.
(a) State the vertex (the deepest point of the reflector) and the axis of symmetry.
(b) Find the depth at $x = 0$ and at $x = 2$. Show that they are equal and explain.
(c) Is the reflector wider or narrower than the standard "$y = x^2$" shape? Justify using $|a|$.
1.3 — Profit from sales. A small bakery models its daily profit by $P = -3(n - 6)^2 + 50$, where $n$ is the number of dozen pastries sold and $P$ is the profit in dollars.
(a) How many dozen should the bakery aim to sell to maximise profit?
(b) What is the maximum profit?
(c) What is the profit at $n = 8$? At $n = 4$? Why are these equal?
1.4 — Roller-coaster loop top. A small section near the top of a roller coaster is shaped like $y = -(x - 3)^2 + 12$, where $x$ is the distance (m) along the track and $y$ is the height (m) above the platform.
(a) What is the highest point of the coaster section and where does it occur?
(b) Does the curve open up or down? Why does that match the physical setting?
(c) How high is the coaster at $x = 1$? Compare with $x = 5$ — what do you notice?
1.5 — Diver's path from a 5 m platform. A diver pushes off a $5$ m platform; her height (m) above the water at horizontal distance $x$ (m) from the platform edge is $y = -2(x - 1)^2 + 7$.
(a) State the maximum height she reaches and how far from the edge it occurs.
(b) What is her starting height ($x = 0$)? Show that this matches the $5$ m platform.
(c) Without solving algebraically, explain why she will definitely hit the water (predict from vertex direction).
2. Explain your thinking
Use full sentences, no dot points. 4 marks
2.1 A classmate is reading the equation $y = 3(x - 2)^2 + 5$. They write: "Vertex is $(-2, 5)$ because we flip the sign of the $-2$, and the parabola is a MAXIMUM at $y = 5$." Write a full-paragraph reply that (i) corrects the vertex's $x$-coordinate AND explains why the rule for $h$ flips, (ii) corrects the min/max claim AND explains how to read whether a vertex is a min or a max from the equation, (iii) gives the correct vertex and the correct min/max claim, and (iv) writes one short check the classmate could do (using $x = h$) to verify the answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Basketball shot
(a) $a = -\tfrac{1}{2}$, $h = 4$, $k = 8$. Vertex $(4, 8)$ — the ball reaches its highest point $8$ m up at a horizontal distance of $4$ m from the shooter. (b) MAX (because $a < 0$ opens down) — matches reality: a thrown ball rises, peaks, then falls. (c) Sub $x = 0$: $y = -\tfrac{1}{2}(0 - 4)^2 + 8 = -\tfrac{1}{2}(16) + 8 = -8 + 8 = \mathbf{0}$ m. (Ball leaves the hand at floor level — possibly a low release point.)
1.2 — Parabolic mirror
(a) Vertex $(1, -3)$ — deepest point of the reflector at $1$ cm along the base, $3$ cm below the rim. Axis $x = 1$. (b) At $x = 0$: $y = 2(-1)^2 - 3 = 2 - 3 = -1$ cm. At $x = 2$: $y = 2(1)^2 - 3 = -1$ cm. Equal because both sit $1$ cm from the axis of symmetry $x = 1$. (c) $|a| = 2 > 1$, so NARROWER than $y = x^2$ — the reflector curves more sharply.
1.3 — Profit from sales
(a) $h = 6$, so sell $\mathbf{6}$ dozen for max profit. (b) $k = 50$, so max profit $= \mathbf{\$50}$. (c) At $n = 8$: $P = -3(8 - 6)^2 + 50 = -3(4) + 50 = 38$. At $n = 4$: $P = -3(-2)^2 + 50 = -12 + 50 = 38$. Both $\$38$. Equal because $n = 8$ and $n = 4$ are both $2$ dozen away from the optimal $n = 6$ — the parabola is symmetric, so equal "distance from optimum" gives equal profit.
1.4 — Roller-coaster loop top
(a) Highest point: vertex $(3, 12)$. Highest height $12$ m at $x = 3$ m along the track. (b) $a = -1 < 0$, so opens DOWN. Matches a coaster's hump — rises to the top, then comes back down. (c) At $x = 1$: $y = -(1 - 3)^2 + 12 = -4 + 12 = 8$ m. At $x = 5$: $y = -(5 - 3)^2 + 12 = -4 + 12 = 8$ m. Equal — symmetric track points 2 m from the peak have the same height.
1.5 — Diver's path
(a) $a = -2$, $h = 1$, $k = 7$. Max height $\mathbf{7}$ m at $x = 1$ m from the platform edge. (b) Sub $x = 0$: $y = -2(0 - 1)^2 + 7 = -2 + 7 = \mathbf{5}$ m. Matches the $5$ m platform height. (c) Vertex $(1, 7)$ sits above the water (positive $k$); curve opens DOWN ($a < 0$). An inverted U starting at $5$ m, peaking at $7$ m, must eventually fall back below zero — that's where she enters the water.
2.1 — Explain your thinking (sample response)
My classmate has two mistakes in one equation. First, the vertex is NOT $(-2, 5)$ — it's $(2, 5)$. The rule for $h$ does flip the sign, but the right way to do it is "zero the bracket": find the $x$ that makes $(x - 2) = 0$, which is $x = 2$ (not $-2$). So the vertex's $x$-coordinate is the OPPOSITE of the sign you see in the bracket — minus inside means positive $h$. Second, the vertex is NOT a maximum — it's a MINIMUM. The min/max comes from the sign of $a$, not from the size of $k$. Here $a = 3 > 0$, so the parabola opens UP and the vertex is the LOWEST point. So the correct reading is: vertex $(2, 5)$, MINIMUM with value $y = 5$. A quick check: substitute $x = h = 2$ into the equation: $y = 3(2 - 2)^2 + 5 = 0 + 5 = 5$. The $y$-value equals $k = 5$, which confirms the vertex is at $(2, 5)$ and the min value really is $5$.
Marking: 1 mark for correct vertex; 1 mark for explaining the "zero the bracket" rule; 1 mark for the min/max correction (sign of $a$); 1 mark for the $x = h$ substitution check.