Question

The curve produced by the water coming from a hose is sketched onto a graph with zeros at 0 and 5. The point (4, 1) also lies on the curve. If h(x) represents the vertical distance from where the water first comes out of the hose and x represents the horizontal distance, which statements are true? Check all that apply. The scenario can be represented by the function h(x) = –0. 25(x)(x – 5). The scenario can be represented by the function h(x) = (x)(x 5). The vertex is on the line x = 2. 5. The greatest height that the water reaches is 1. 5 units. The scenario can be represented by the function h(x) = –1(x – 5).

Answer 1

You can use the fact that at the zeros of a function, the function evaluates to 0.

For the given scenario, the following statements are correct:

Option A: The scenario can be represented by the function h(x) = -0.25(x)(x-5)

Option B: The vertex is on the line x = 2.5

How to know which function has zeros where?

If function can be written in all factored form with form as (x+a), then we can one by one equate them to zero to get the zeros of given function. If not, then we will have to find them by other methods.

How to form the quadratic equation representing the given scenario?

Since the given scenario has zeros at 0 and 5, thus two factors of the function h(x) would be (x-0)(x-5) = (x)(x-5). Since at x = 4, the function outputs 1, thus we have: h(x) = a(x)(x-5)( here a is other factor with two already known factors of function h(x) which might represent the scenario (assuming that scenario given can be represented or modeled using quadratic equation) such that:

a(4)(4-5) = 1,

or

$a(4)(4-5) = 5\\ a(4)(-1) = 5\\ a = -\dfrac{5}{4} = -0.25$

Thus, the function for given scenario is represented by

$h(x) = a(x)(x-5) = -0.25(x)(x-5)$

Finding the first and second rate of function to get the maxima or minima:

$h(x) = -0.25(x)(x-5) = -0.25x^2 + 1.25x\\ h'(x) = -0.5x + 1.25\\ h''(x) = -0.5$

Putting h'(x) = 0 gives:

$-0.5x + 1.25 = 0\\\\ x = \dfrac{1.25}{0.5} = 0.25$

For all x, the second rate is negative, showing that at x = 2.5, there is maxima. Thus the vertex is on the line x =2.5 and the greatest height water achieves is at x = 2.5

Thus,

Option A: The scenario can be represented by the function h(x) = -0.25(x)(x-5)

Option B: The vertex is on the line x = 2.5; are correct options.

Learn more about zeros of quadratic equations here:

https://brainly.com/question/141791

Answer 2

Answer:

A&C

Step-by-step explanation: