# A bottle rocket travels along a parabolic path and reaches a maximum height of 21 feet after traveling a horizontal distance of 7 feet.

A) Write a quadratic function of the form y=a(x-h)^2 +k that models the bottle rocket's path, assuming it leaves the ground at the point (0,0).

B) Describe how changing the values of a, h, and k affect the flight path of the bottle rocket.

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for A) a classmate got y=-3/7 (x-7)^2 +21 ... I was wondering how they got that as their answer ... Oh wait so they divided 7 by 21 to get the 3 ?

## the vertex of your parabolic path is (7,21)

so y = a(x-7)^2 + 21
but (0,0) lies on it, so
0 = a(0-7)^2 + 21
0 = 49a + 21
49a = -21
a = -21/49 = -3/7

so y = (-3/7)(x - 7)^2 + 21

## Well, your classmate's answer is close, but there seems to be a slight mistake. Let's break it down step by step:

A) To find the quadratic function that models the bottle rocket's path, we first need to determine the values of a, h, and k in the general form equation y = a(x - h)^2 + k.

We know that the rocket reaches a maximum height of 21 feet, so the vertex of the parabolic path is (h, k) = (7, 21). Therefore, h = 7 and k = 21.

Now, we just need to find the value of a. We can use the point (0, 0) as another point on the parabola.

Plugging in these values, we get:
0 = a(0 - 7)^2 + 21
0 = 49a + 21
49a = -21
a = -21/49 = -3/7

So, the correct quadratic function that models the bottle rocket's path is y = -3/7(x - 7)^2 + 21.

B) Now, let's discuss how changing the values of a, h, and k affect the flight path of the bottle rocket.

- The value of a determines the shape and direction of the parabola. A positive value will open the parabola upwards (like the bottle rocket's path) while a negative value will open it downwards. Changing the value of a will affect how steep or shallow the parabola is.

- The value of h represents the horizontal shift of the parabola. Changing h will move the entire parabola left or right. In the case of the bottle rocket, h = 7 means that the vertex (highest point) of the parabola occurs at x = 7.

- The value of k indicates the vertical shift of the parabola. Changing k will move the entire parabola up or down. In the case of the bottle rocket, k = 21 means that the vertex (highest point) is at the height of 21 feet.

So, changing the values of a, h, and k will affect the shape, position, and orientation of the parabolic flight path of the bottle rocket.

## No, your classmate did not divide 7 by 21 to get 3. Let me explain how they derived the quadratic function and why it is correct.

A) To write a quadratic function in the form y=a(x-h)^2 + k that models the bottle rocket's path, we need to determine the values of a, h, and k based on the given information.

Given:
Maximum height = 21 feet
Horizontal distance traveled = 7 feet

Since the maximum height occurs at the vertex of the parabolic path, the vertex form of the quadratic function can be used.

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) represents the vertex.

From the given data, we know that the vertex is (7, 21).

To find the value of a, we can use either the vertex form or substitute a point on the parabola into the equation.

Since the rocket starts at the point (0, 0), we can substitute these values into the equation:

0 = a(0 - 7)^2 + 21

Simplifying the equation, we get:

0 = a(49) + 21

0 = 49a + 21

Subtracting 21 from both sides, we get:

-21 = 49a

Dividing both sides by 49, we get:

a = -21/49 = -3/7

Therefore, the quadratic function that models the bottle rocket's path is:

y = (-3/7)(x - 7)^2 + 21

B) Changing the values of a, h, and k will affect the flight path of the bottle rocket as follows:

1. Parameter 'a':
- Positive values of 'a' will make the parabola open upwards, while negative values will make it open downwards.
- Larger absolute values of 'a' will make the parabola steeper.
- A value of 'a' equal to zero will create a linear function.

2. Parameter 'h':
- Changing the value of 'h' shifts the vertex of the parabola horizontally.
- If 'h' is positive, the vertex moves to the right, and if 'h' is negative, it moves to the left.

3. Parameter 'k':
- Changing the value of 'k' shifts the vertex of the parabola vertically.
- If 'k' is positive, the vertex moves upward, and if 'k' is negative, it moves downward.

In summary, changing 'a' affects the shape and steepness of the parabola, changing 'h' shifts the parabola horizontally, and changing 'k' shifts the parabola vertically.

## To find the equation in the form y = a(x-h)^2 + k that models the bottle rocket's path, we need to use the information given in the problem.

First, let's analyze the information provided. The bottle rocket reaches its maximum height of 21 feet after traveling a horizontal distance of 7 feet. We know that the vertex of the parabolic path will be at the point (h, k), where h represents the horizontal distance traveled and k represents the maximum height reached.

Now, let's break down the equation step by step:

1. The vertex forms the equation as (h,k), which means the maximum height of the parabolic path is at the point (7, 21).

2. The equation is of the form y = a(x-h)^2 + k, where (h, k) is the vertex. Plugging in the values (7, 21), we get y = a(x-7)^2 + 21.

3. To find the value of a, we need one more point on the parabolic path. Since the bottle rocket leaves the ground at the point (0, 0), we can substitute these values into the equation: 0 = a(0-7)^2 + 21.

Simplifying this equation:

0 = a(-7)^2 + 21
0 = 49a + 21
-21 = 49a
a = -21/49
a = -3/7

Hence, the correct equation that models the bottle rocket's path is y = -3/7(x-7)^2 + 21.

Moving on to part B) of the question, let's discuss how changing the values of a, h, and k affect the flight path of the bottle rocket:

1. The value of 'a' determines the steepness or the opening direction of the parabola. If 'a' is positive, the parabola opens upwards, indicating that the rocket ascends and then descends. Conversely, if 'a' is negative, the parabola opens downwards.

2. The value of 'h' represents the horizontal translation or the horizontal distance the rocket has traveled. Changing 'h' shifts the entire parabolic path left or right. A positive 'h' value shifts the parabola to the right, while a negative 'h' value shifts it to the left.

3. The value of 'k' represents the vertical translation or the maximum height reached by the bottle rocket. Altering 'k' shifts the parabolic path up or down. A positive 'k' value shifts the parabola upward, while a negative 'k' value shifts it downward.

In summary, changing the value of 'a' affects the shape of the flight path, 'h' determines the horizontal position of the vertex, and 'k' changes the maximum height of the rocket's trajectory.