The shape of an arch

The shape of an arch

One quadratic, read four ways: as zeros, as a vertex, as a constraint, and as an equivalent expression.

1. 12.1Quadratic functionsFoundation

   Where does the arch meet the deck?Where does the arch meet the deck?

   • AAt x = 0 and x = 20At x = 0 and x = 20
   • BAt x = 0 and x = 16At x = 0 and x = 16
   • CAt x = 0 onlyAt x = 0 only
   • DAt x = 1.6 and x = 20At x = 1.6 and x = 20
   Show the answer and reasoningHide the reasoning

   At x = 0 and x = 20At x = 0 and x = 20

   Set h(x) = 0 and factor: x(-0.08x + 1.6) = 0, so x = 0 or -0.08x = -1.6, giving x = 20. The arch spans 20 meters.Set h(x) = 0 and factor: x(-0.08x + 1.6) = 0, so x = 0 or -0.08x = -1.6, giving x = 20. The arch spans 20 meters.

2. 12.2Quadratic functionsStandard

   Builds on: The zeros from question 1. Without them you would reach for the formula x = -b / 2a; with them the midpoint is quicker and harder to get wrong.The zeros from question 1. Without them you would reach for the formula x = -b / 2a; with them the midpoint is quicker and harder to get wrong.

   How high is the arch at its highest point, and where is that point?How high is the arch at its highest point, and where is that point?

   • A8 meters, 10 meters from the left base8 meters, 10 meters from the left base
   • B16 meters, 10 meters from the left base16 meters, 10 meters from the left base
   • C8 meters, 8 meters from the left base8 meters, 8 meters from the left base
   • D1.6 meters, 20 meters from the left base1.6 meters, 20 meters from the left base
   Show the answer and reasoningHide the reasoning

   8 meters, 10 meters from the left base8 meters, 10 meters from the left base

   A parabola is symmetric about its vertex, so the maximum sits midway between the zeros, at x = 10. Then h(10) = -0.08(100) + 16 = -8 + 16 = 8 meters.A parabola is symmetric about its vertex, so the maximum sits midway between the zeros, at x = 10. Then h(10) = -0.08(100) + 16 = -8 + 16 = 8 meters.

3. 12.3Interpreting modelsChallenge

   Builds on: Question 1 fixed the span, which is what "centered" means here, and question 2 supplied the symmetry. The third choice is the trap: clearance is decided at the tightest point, not the peak.Question 1 fixed the span, which is what "centered" means here, and question 2 supplied the symmetry. The third choice is the trap: clearance is decided at the tightest point, not the peak.

   A truck 4 meters tall and 12 meters wide is driven under the arch, centered on the span. Does it pass?A truck 4 meters tall and 12 meters wide is driven under the arch, centered on the span. Does it pass?

   • AYes — the tightest point is 5.12 meters, leaving 1.12 meters of clearance.Yes — the tightest point is 5.12 meters, leaving 1.12 meters of clearance.
   • BNo — the arch is lower than 4 meters at the edges of the truck.No — the arch is lower than 4 meters at the edges of the truck.
   • CYes, because the arch reaches 8 meters at its peak.Yes, because the arch reaches 8 meters at its peak.
   • DIt cannot be decided without knowing how long the truck is.It cannot be decided without knowing how long the truck is.
   Show the answer and reasoningHide the reasoning

   Yes — the tightest point is 5.12 meters, leaving 1.12 meters of clearance.Yes — the tightest point is 5.12 meters, leaving 1.12 meters of clearance.

   Centered on a 20-meter span, a 12-meter truck runs from x = 4 to x = 16. The arch is lowest over that stretch at its two edges: h(4) = -1.28 + 6.4 = 5.12 meters, and h(16) is the same by symmetry. Since 5.12 > 4, it passes.Centered on a 20-meter span, a 12-meter truck runs from x = 4 to x = 16. The arch is lowest over that stretch at its two edges: h(4) = -1.28 + 6.4 = 5.12 meters, and h(16) is the same by symmetry. Since 5.12 > 4, it passes.

4. 12.4Equivalent expressionsChallenge

   Builds on: The vertex (10, 8) from question 2 — vertex form is that pair of numbers written into the expression, which is why reading it off is faster than expanding all four choices.The vertex (10, 8) from question 2 — vertex form is that pair of numbers written into the expression, which is why reading it off is faster than expanding all four choices.

   Which expression is equivalent to h(x) and shows the maximum height as a constant?Which expression is equivalent to h(x) and shows the maximum height as a constant?

   • Ah(x) = -0.08(x - 10)² + 8h(x) = -0.08(x - 10)^2 + 8
   • Bh(x) = -0.08(x - 8)² + 10h(x) = -0.08(x - 8)^2 + 10
   • Ch(x) = -0.08x(x - 20)h(x) = -0.08x(x - 20)
   • Dh(x) = -0.08(x + 10)² + 8h(x) = -0.08(x + 10)^2 + 8
   Show the answer and reasoningHide the reasoning

   h(x) = -0.08(x - 10)² + 8h(x) = -0.08(x - 10)^2 + 8

   Expanding gives -0.08(x² - 20x + 100) + 8 = -0.08x² + 1.6x - 8 + 8, which is h(x). The second choice swaps the two vertex coordinates and gives 9.68 rather than 8 at x = 10. The third is equivalent but factored, so it shows the zeros instead. The fourth puts the vertex at x = -10.Expanding gives -0.08(x^2 - 20x + 100) + 8 = -0.08x^2 + 1.6x - 8 + 8, which is h(x). The second choice swaps the two vertex coordinates and gives 9.68 rather than 8 at x = 10. The third is equivalent but factored, so it shows the zeros instead. The fourth puts the vertex at x = -10.