Fountains The height of a fountain's water stream can be...

Fountains The height of a fountain's water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet. Suppose a worker increases the water pressure so that the stream reaches a maximum height of 12.5 feet at a distance of 15 inches from the jet. The water now lands 3.75 feet from the jet. Write a new quadratic function for $H(d) .$ How do the changes in $h$ and $k$ affect the shape of the graph?

Fountains The height of a fountain's water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet.
Suppose a worker increases the water pressure so that the stream reaches a maximum height of 12.5 feet at a distance of 15 inches from the jet. The water now lands 3.75 feet from the jet. Write a new quadratic function for $H(d) .$ How do the changes in $h$ and $k$ affect the shape of the graph?

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Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form f(x) = ax² + bx + c, which graph as parabolas. They are commonly used to model trajectories and other phenomena where the relationship between variables is curved. The quadratic function’s key feature is its symmetry, with the vertex representing the maximum or minimum point depending on the sign of the coefficient a.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is especially useful when modeling physical situations since the vertex corresponds to the peak (or lowest) value of the function. In contexts such as projectile motion or water fountains, it easily conveys the maximum height and the horizontal position where that height is reached.

Graph Transformations via h and k

Changing the values of h and k in the vertex form of a quadratic function results in horizontal and vertical translations, respectively. The parameter h shifts the parabola left or right, altering the position of the vertex along the x-axis, while the parameter k moves the parabola up or down, changing the maximum (or minimum) value. These transformations affect the location of the graph without altering its overall shape, which is determined by the coefficient a.

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