Mathematics • Year 9 • Unit 2 • Lesson 7
Sliding Parabolas in Real Settings
Five scenarios where the parabola has been shifted sideways: a roller coaster dip, a skateboard half-pipe, a goal-kick distance check, a paddock fence design, and a hockey-stick logo. Read $h$ off each.
1. Word problems
For each scenario: identify $h$, find the requested features, and answer in plain English. 3 marks each
1.1 — Roller-coaster dip. A small section of a roller coaster is modelled by $y = (x - 5)^2$, where $x$ is the horizontal distance (m) along the track and $y$ is the height (m) above the lowest point.
(a) Where (at what $x$-value) is the lowest point of this section?
(b) What is the height of the track at $x = 0$ (the start of this section)?
(c) The carriage is symmetrical about the dip. At what other $x$-value does the height match the value at $x = 0$?
1.2 — Skateboard half-pipe. The cross-section of a skateboard half-pipe is approximately $y = (x - 3)^2$, where $x$ is the distance (m) from the left wall and $y$ is the height (m) above the lowest point of the floor.
(a) State the vertex and what it represents physically.
(b) Find the height of the wall at $x = 0$ (left edge) and at $x = 6$ (right edge).
(c) Show that the heights in (b) are equal, and explain why this is expected.
1.3 — Goal-kick distance check. A goal kick is modelled by $y = (x - 4)^2$, where $x$ is the horizontal distance (m) from the kick spot and $y$ is the squared distance from the goal posts (a "distance score"). The goal sits at the vertex.
(a) Where (at what $x$-value) is the goal located?
(b) Compute the "distance score" at $x = 0$ and $x = 8$. Which is closer to the goal?
(c) A defender starts at $x = 1$ and an attacker starts at $x = 7$. Which one is closer to the goal? Justify using the equation.
1.4 — Paddock fence design. A farmer is designing a curved fence with cross-section $y = (x + 2)^2$, where $x$ is the position (m) along the paddock and $y$ is the height (m) of the fence above ground.
(a) At what $x$-value is the fence at ground level ($y = 0$)?
(b) How tall is the fence at $x = 0$ (the entry gate)?
(c) How tall is the fence at $x = -4$? Compare with (b) — what does this tell you about the symmetry?
1.5 — Logo design. A hockey-stick logo uses a parabola with the SAME shape as $y = x^2$ but with its vertex slid 6 units to the RIGHT.
(a) Write the equation of the logo's parabola.
(b) State the vertex.
(c) Find where the logo crosses the $y$-axis (the $y$-intercept).
2. Explain your thinking
Use full sentences, no dot points. 4 marks
2.1 A friend looks at $y = (x - 3)^2$ and says "the minus 3 means the curve has slid 3 to the LEFT — minus means left, plus means right, just like on a number line." Write a full-paragraph response that (i) explicitly says which direction the curve has actually shifted, (ii) explains the test that finds the vertex (the "zero the bracket" idea), (iii) shows the same test applied to $y = (x + 5)^2$ to confirm it ends up on the LEFT, and (iv) gives your friend a one-line memory rule to avoid this mistake in future.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Roller-coaster dip
(a) Lowest point where the bracket is zero: $x - 5 = 0 \Rightarrow x = 5$. So at $\mathbf{x = 5}$ m. (b) At $x = 0$: $y = (0 - 5)^2 = 25$ m. (c) Axis of symmetry is $x = 5$; the start ($x = 0$) is 5 units to the left of the axis, so the matching point is 5 units to the right at $\mathbf{x = 10}$ m. (Check: $y = (10 - 5)^2 = 25$ m — matches.)
1.2 — Skateboard half-pipe
(a) Vertex $(3, 0)$ — this is the lowest point of the floor, 3 m from the left wall. (b) Left wall ($x = 0$): $y = (0 - 3)^2 = 9$ m. Right wall ($x = 6$): $y = (6 - 3)^2 = 9$ m. (c) Both heights equal 9 m, which is expected because the walls are symmetric about the floor's lowest point — they sit equal distances (3 m) from the axis of symmetry $x = 3$.
1.3 — Goal-kick distance check
(a) Goal at the vertex: $x - 4 = 0 \Rightarrow \mathbf{x = 4}$. (b) At $x = 0$: $y = 16$. At $x = 8$: $y = 16$. Both have the same "distance score" — both are 4 m from the goal, just on opposite sides. (c) Defender at $x = 1$: $y = (1 - 4)^2 = 9$. Attacker at $x = 7$: $y = (7 - 4)^2 = 9$. They are EQUALLY close to the goal — both 3 m away. The attacker is not closer just because they have a bigger $x$; what matters is distance from the axis $x = 4$.
1.4 — Paddock fence design
(a) $y = 0$ when $(x + 2)^2 = 0 \Rightarrow x = -2$ m. (b) At $x = 0$: $y = (0 + 2)^2 = 4$ m tall. (c) At $x = -4$: $y = (-4 + 2)^2 = (-2)^2 = 4$ m tall. Same height as (b), which makes sense because both points sit 2 m from the axis of symmetry $x = -2$ — the fence is mirror-symmetric about its lowest point.
1.5 — Logo design
(a) Shift right by 6 $\Rightarrow$ $h = +6$ $\Rightarrow$ equation $y = (x - 6)^2$. (b) Vertex $(6, 0)$. (c) $y$-intercept: sub $x = 0$: $y = (0 - 6)^2 = 36$, so $(0, 36)$.
2.1 — Explain your thinking (sample response)
My friend is wrong — $y = (x - 3)^2$ has actually shifted 3 units to the RIGHT, not to the left, even though there's a minus sign inside the bracket. The test that finds the vertex is "zero the bracket": ask "what $x$ makes the inside equal zero?" For $(x - 3)$, the bracket is zero when $x = 3$, so the vertex sits at $(3, 0)$ — that's 3 to the right of the origin. The opposite of what the minus sign suggests. Now apply the same test to $y = (x + 5)^2$: the bracket is zero when $x = -5$, so the vertex is at $(-5, 0)$ — 5 to the LEFT of the origin. So PLUS inside means LEFT, MINUS inside means RIGHT. My friend has the rule exactly backwards. One-line memory rule: "Whatever sign you SEE inside the bracket, the vertex sits on the OPPOSITE side of zero."
Marking: 1 mark for stating the correct direction (right); 1 mark for the "zero the bracket" test; 1 mark for confirming with $(x + 5)^2$; 1 mark for the memory rule.