Exercise 81 Page 85

The function will look a little different depending on where in the coordinate plane you choose to place the graph. Let's choose Maura's feet to be at the origin. Since the nozzle is $4$ feet above the ground, its coordinates will be $(0, 4) .$

The top of the flowers is at the same height as the nozzle, but $10$ feet to the right.

The highest point is $8$ feet off the ground, and since the model that Maura's using is a parabola it's in between the flowers and nozzle, giving it the coordinates $(5, 8) .$

This parabola can be described by a quadratic function.

y = a (x - h)^{2} + k

This is the so-called vertex form of a quadratic function. Here,

(h, k)

is the vertex, and since the maximum point is

(5, 8)

that gives us

h = 5

and

k = 8 .

y = a (x - 5)^{2} + 8

To find the coefficient

a,

let's use a point that we know is on the graph,

(0, 4) .

y = a (x - 5)^{2} + 8

SubstituteII

$x = 0$ , $y = 4$

4 = a (0 - 5)^{2} + 8

▼

Solve for

a

SubTerm

Subtract term

4 = a (- 5)^{2} + 8

NegBaseToPosBase

$(- a)^{2} = a^{2}$

4 = a (25) + 8

SubEqn

$LHS - 8 = RHS - 8$

- 4 = a \cdot 25

RearrangeEqn

Rearrange equation

a \cdot 25 = - 4

DivEqn

$LHS / 25 = RHS / 25$

a = - \frac{4}{2 5}

The coefficient is

a = - \frac{4}{2 5},

so let's put that in the function.

y = - \frac{4}{2 5} (x - 5)^{2} + 8

Domain and Range

What is the domain and range? The domain is what

x -

values we can substitute into the function. The parabola is a good approximation of the water's journey between the nozzle and the flowers, so we can write a reasonable domain.

0 \leq x \leq 10

In this domain, the height of the water varies between

4

and

8

feet including the endpoints. With this information, we can write the range.

4 \leq y \leq 8