Water Trajectory: Finding The Curve Equation H(x)

by Luna Greco 50 views

Have you ever noticed the graceful arc of water as it streams out of a garden hose? It's a familiar sight, but have you ever thought about the mathematics behind it? The curve formed by the water is a parabola, a U-shaped curve that's described by a quadratic equation. Let's dive into the fascinating world of parabolas and explore how we can use mathematical principles to understand the trajectory of water from a hose.

Visualizing the Water's Path: A Parabola in Action

To begin our exploration, imagine sketching the path of the water onto a graph. We'll place the starting point of the water stream at the origin (0, 0) of our coordinate system. As the water flows, it rises to a certain height and then falls back down, eventually hitting the ground. Let's say the water lands 5 units away from the starting point. This gives us another point on our graph: (5, 0). These two points, (0, 0) and (5, 0), are known as the zeros or x-intercepts of the parabola. They are the points where the curve intersects the x-axis.

Now, let's add another piece of information. Suppose we know that the point (4, 1) also lies on the curve. This means that when the horizontal distance from the starting point is 4 units, the vertical distance (or height) of the water is 1 unit. This additional point will be crucial in helping us determine the specific equation of the parabola.

The General Form of a Parabola

Before we get into the specifics of our water trajectory, let's revisit the general form of a quadratic equation, which describes a parabola:

y = ax² + bx + c

Here, a, b, and c are constants that determine the shape and position of the parabola. The coefficient a plays a particularly important role: it determines whether the parabola opens upwards (if a is positive) or downwards (if a is negative). Since the water stream forms an upside-down U shape, we know that a must be negative in our case.

Harnessing the Zeros: A Simplified Form

Knowing the zeros of the parabola can significantly simplify the equation. If we know the zeros are r and s, we can write the equation in the following factored form:

y = a(x - r)(x - s)

This form is incredibly useful because it directly incorporates the zeros into the equation. In our case, the zeros are 0 and 5, so we can write the equation as:

y = a(x - 0)(x - 5)

Simplifying this, we get:

y = ax(x - 5)

Now, we only need to find the value of a to completely determine the equation of our parabola. This is where the point (4, 1) comes in handy.

Finding the Missing Piece: Using the Point (4, 1)

We know that the point (4, 1) lies on the curve, which means that when x is 4, y is 1. We can substitute these values into our equation to solve for a:

1 = a(4)(4 - 5)

Simplifying, we get:

1 = a(4)(-1)
1 = -4a

Dividing both sides by -4, we find:

a = -1/4

So, the value of a is -1/4. Now we have all the pieces we need to write the complete equation of the parabola.

The Equation of the Water's Trajectory

Substituting the value of a back into our equation, we get:

y = (-1/4)x(x - 5)

This is the equation that describes the parabolic trajectory of the water from the hose. We can also expand this equation to get it into the standard quadratic form:

y = (-1/4)x² + (5/4)x

This equation, in either factored or standard form, allows us to predict the height of the water at any horizontal distance from the starting point.

Representing the Vertical Distance: h(x)

In the context of the problem, we're given that h(x) represents the vertical distance from where the water exits the hose. This is simply another way of saying that h(x) is the y-coordinate of the parabola at a given x-coordinate. So, we can write:

h(x) = (-1/4)x(x - 5)

Or equivalently:

h(x) = (-1/4)x² + (5/4)x

This notation emphasizes that the vertical distance, h, is a function of the horizontal distance, x. For any value of x, we can plug it into this equation to find the corresponding height of the water stream.

Guys, Let's Break Down the Key Concepts

Okay, so we've covered a lot of ground here. Let's recap the main ideas in a way that's super clear and easy to remember:

  • Parabolas and Quadratic Equations: The path of water from a hose is a parabola, which is described by a quadratic equation (y = ax² + bx + c). Think of it like this: the water's going on a quadratic adventure!
  • Zeros (x-intercepts): These are the points where the parabola crosses the x-axis (y = 0). They're like the starting and ending points of our water's journey.
  • Factored Form: Knowing the zeros lets us write the equation in a simpler form: y = a(x - r)(x - s), where r and s are the zeros. It's like having a secret code to unlock the parabola's equation!
  • Using an Additional Point: One extra point on the curve (like our (4, 1)) helps us find the value of 'a', which determines the parabola's shape. This is like finding the missing ingredient in our parabola recipe.
  • h(x) Notation: This just means the vertical distance (height) is a function of the horizontal distance. It's a fancy way of saying