Daredevil Danny takes a practice jump as shown on the graph below. Complete the steps to determine the equation for the parabolic trajectory of his practice jump.
What are the key aspects of any parabola? What do the key aspects tell you about the graph?
(5 points)
What are the key aspects of Daredevil Danny’s practice jump shown in the graph provided? Describe each value and how it is determined. (10 points)
Step 2: Solve for coefficient a.
What equation is used to find the vertex form of a parabola with the vertex (h, k)? (5 points)
Solve for the coefficient a in the vertex form for the parabola in Daredevil Danny’s practice jump using key aspects from step 1. As you solve for the coefficient, explain each step of the process in detail. (15 points)
Flaming Hoop Jump of Awesome Daredevil Danny attempts the feat of jumping his motorcycle through the Flaming Hoop Jump of Awesome. In order for Daredevil Danny to pass through the hoop, he will need a safe path to travel. Let’s explore the parabolic trajectory that he will need for a safe journey. Directions Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated, including inserting images of graphs. Be sure that all graphs or screenshots include appropriate information such as titles, labeled axes, etc. If your word processing program has an equation editor, you can insert your equations here. Otherwise, print this activity sheet and write your answers by hand. In addition to the answers you determine, you will be graded based on the work you show, or your solution process. So, be sure to show all your work and answer each question as you complete the task. Type all your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations. Your teacher will give you further directions about how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or print it and hand in a hard copy. Now, let’s get started! Step 1: Analyze the graph. Daredevil Danny takes a practice jump as shown on the graph below. Complete the steps to determine the equation for the parabolic trajectory of his practice jump. What are the key aspects of any parabola? What do the key aspects tell you about the graph? (5 points) What are the key aspects of Daredevil Danny’s practice jump shown in the graph provided? Describe each value and how it is determined. (10 points) Step 2: Solve for coefficient a. What equation is used to find the vertex form of a parabola with the vertex (h, k)? (5 points) Solve for the coefficient a in the vertex form for the parabola in…
The Correct Answer and Explanation is :
Step 1: Key Aspects of Any Parabola
A parabola is defined by its symmetrical curve and can be expressed using a quadratic equation of the form:
[ y = a(x – h)^2 + k ]
where:
- (h, k) is the vertex of the parabola, the highest or lowest point on the graph depending on the direction it opens.
- a determines the “width” and direction of the parabola (whether it opens upward or downward).
- The axis of symmetry passes through the vertex and divides the parabola into two symmetrical halves.
- The focus and directrix are geometric properties related to the parabola’s definition but are not always visible on the graph.
These aspects are crucial in determining the trajectory or path of an object moving in a parabolic path, such as Daredevil Danny’s motorcycle jump. The key aspects will help us identify important points like the maximum height (vertex) and the points where the jump touches the ground.
Key Aspects of Daredevil Danny’s Practice Jump (as shown in the graph)
To describe the key aspects of Daredevil Danny’s practice jump, you would need to identify:
- Vertex (h, k): The highest point of his jump (the vertex of the parabola). This point can be identified from the graph as the maximum y-value, which corresponds to Danny’s highest point in the jump.
- Initial height: This is the y-coordinate when the motorcycle first leaves the ground (often where x = 0).
- Landing point: The x-coordinate where the jump touches the ground again, often where y = 0.
- Direction of the parabola: Is the parabola opening upward (suggesting the motorcycle goes up and then comes back down) or downward (suggesting a possible fall trajectory or a different jump type)?
These aspects can be visually extracted from the graph. The vertex represents the peak, and the width and direction of the parabola provide information about the jump’s distance and height.
Step 2: Solve for Coefficient a
The vertex form of the equation for a parabola is given as:
[ y = a(x – h)^2 + k ]
Where (h, k) is the vertex. To solve for the coefficient a, we can use another point on the parabola, preferably one where we know the value of x and y. This is typically a point where the object is at ground level, i.e., y = 0.
Equation to Solve for a:
We can plug in the known values for x, y, and the vertex (h, k) into the equation to solve for a. For example, suppose we know that the jump touches the ground at x = x₁ (where y = 0), we would substitute into the equation:
[ 0 = a(x₁ – h)^2 + k ]
Solving for a:
[
a = \frac{-k}{(x₁ – h)^2}
]
Explanation:
- From the graph, the vertex (h, k) will give us the location of the highest point of the jump.
- By using a point (x₁, 0) where the motorcycle lands, we can substitute these values into the equation to find the coefficient a.
- This step is important because a determines the curvature of the parabola, so solving for it allows us to fully define the equation for the jump’s trajectory.
Each of these steps contributes to constructing the full quadratic equation that models Daredevil Danny’s jump. The final equation will allow us to predict the trajectory of the motorcycle through the Flaming Hoop Jump of Awesome.