A parabolic arch with a stated peak gives the vertex, the curve’s turning point, so start with vertex form. The peak fixes where the arch is highest, but not how quickly it falls. Use a second point to find the coefficient of the squared term, with a one-unknown Desmos regression. Don’t mistake an off-center height for the height at the center.
Hints
- Hint 1
The arch’s highest point is its vertex. In vertex form, , the vertex is . What do the center’s input and the peak’s height tell you about and ?
- Hint 2
A point on a graph pairs an input with an output. Use the height away from the center to write , then substitute it into your equation. This point can tell you the value of .
- Hint 3
In Desmos, replace with in that one-unknown equation. A regression finds the coefficient that makes both sides match. Its sign should be negative because the arch falls away from its peak.
Step-by-step
Start with the vertex, then find the coefficient
Step 1Build the equation from the peak
The highest point is feet high at , so the vertex is . In vertex form, has vertex , giving
Because is the maximum height, must be negative: away from , the squared term must lower the height.
- Step 2
Turn the other height into an equation
Use for the point feet from the center. Its height is feet, so . Substitute this off-center point into the model:
The vertex itself couldn’t determine : at , the squared term would be .
- Step 3
Find the coefficient in Desmos
Type . The regression symbol tells Desmos to find the value of that fits the equation. Under PARAMETERS, Desmos shows , which also has the negative sign a peak requires.
- Step 4
Write the arch’s equation
Replace with :
This model peaks at 6 feet above the ground at the center, and the regression ensures its height at is feet. Choice A.