A quadratic arch with a highest point points to vertex form: its squared term measures horizontal distance from the peak. Use another point to find how quickly the height drops, then define and evaluate the function in Desmos. Treating the rise to the peak as a straight line misses the arch’s curve.
Hints
- Hint 1
The vertex is the parabola’s turning point. At the highest point , the squared part of vertex form must be zero when . What expression involving becomes zero there?
- Hint 2
A point on the graph means input gives output . Use the other point, , to make an equation for the unknown coefficient. The highest point itself cannot determine that coefficient.
- Hint 3
Once you find the coefficient, define the function in Desmos and enter . The requested input is one foot from the left end, even though the square in vertex form measures distance from the peak.
Step-by-step
Build the arch from its highest point
Step 1Write vertex form
The vertex is the parabola’s turning point, so the highest point gives vertex form:
Here controls how quickly the height changes. The square measures distance from the peak, not from the left end.
- Step 2
Use a different point to find the missing coefficient
The point means input gives height , so . Substitute into the form you wrote:
Use this other point: at the vertex, the square is zero for every value of .
- Step 3
Find the coefficient in Desmos
Type . The regression symbol asks Desmos to find ; under PARAMETERS, it shows . A negative coefficient makes the arch open downward, as a highest point requires.
- Step 4
Define the height function
Type . Desmos uses the stored value , so this defines for the input you need next.
- Step 5
Find the height one foot from the left end
Enter , and Desmos shows . The input is because the location is one foot from the left end; the arch is feet tall there. Choice B.