Quadratic equations: will the fountain reach the balcony?
Shape a fountain jet y = −x² + bx + c and find where it reaches a balcony, or prove with the discriminant that it never can.
Core concept
Solving −x² + bx + c = h finds where a parabola meets a line; the discriminant tells whether there are two, one or no real solutions
Topic: Quadratic equations and graphs
The water from a fountain nozzle follows a curve called a parabola. Measured in metres, its height y at a distance x is y = −x² + bx + c: c is the height of the nozzle and b is how hard the pump pushes. A balcony at height h is the line y = h. The jet reaches the balcony where −x² + bx + c = h, which rearranges to the quadratic equation x² − bx + (h − c) = 0. Its discriminant D = b² − 4(h − c) tells us what happens: if D > 0 the water passes height h twice (on the way up and on the way down), if D = 0 it just touches it at its highest point, and if D < 0 it never gets that high.
What you will learn
- Read the graph of a quadratic function: its highest point and where it meets a line.
- Rearrange a problem into a quadratic equation and solve it with the formula.
- Use the discriminant to decide how many real solutions there are.
Before you start
- Squares, square roots and substituting into a formula.
Materials
- A garden hose or a plastic bottle with a small hole
Equipment
- A measuring tape
Procedure
- Tap − or + to change the pump (b); drag the nozzle up or down to change c.
- Drag the purple balcony line to a height h.
- Read the equation, the discriminant and the crossing points.
- Record where the jet lands on the ground, a jet that just touches the balcony, one that never reaches it, and one that passes a balcony 5 m or higher twice.
How this simulation is simplified
- Lengths are in metres; air resistance is ignored.