4Flaming 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. a) What are the key aspects of any parabola? What do the key aspects tell you about the graph? (5 points) b) 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. a) What equation is used to find the vertex form of a parabola with the vertex (h, k)? (5 points) b) 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) c) What is the vertex form of the parabola from Daredevil Danny’s practice jump? (5 points) Step 3: Change from vertex form to standard form. To get Daredevil Danny through the Flaming Hoop Jump of Awesome, you will need to know the values of a, b, and c, which are the coefficients of the quadratic equation in standard form. a) What is the equation for a parabola in standard form? (5 points) b) Describe how to convert from vertex form to standard form. (10 points c) Convert the vertex form of the equation you wrote above to standard form. (10 points) d) What are the values for coefficients a, b, and c in the quadratic equation for Daredevil Danny’s practice jump? (5 points) Step 4: Calculate the equation for the flaming hoop jump. Using the same method for the practice jump, write the standard form of the equation that will get Daredevil Danny safely through the Flaming Hoop Jump of Awesome! a) Write the vertex form by substituting in values for x, y, h, and k, and then solve for a. Show your work. (10 points) b) Determine the values of a, b, and c. Show your work. (10 points) c) Using your values of a, b, and c, did Daredevil Danny successfully pass through the Flaming Hoop Jump of Awesome? If not, describe what happened. Then, go back to critique your work and revise it. Write about your revision. (5 points) d) Once you have determined the correct values of a, b, and c, verify algebraically that Daredevil Danny will successfully pass through the Flaming Hoop Jump of Awesome.
The Correct Answer and Explanation is :
To help you complete the project on Daredevil Danny’s jump through the Flaming Hoop Jump of Awesome, let’s go through each step with an organized approach. Below, I will outline the necessary calculations, descriptions, and explanations required for each step.
Step 1: Analyze the Graph
a) Key Aspects of Any Parabola
The key aspects of a parabola include:
- Vertex (h, k): The highest or lowest point of the parabola, indicating the maximum height of the jump.
- Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
- Y-intercept: The point where the parabola intersects the y-axis, indicating the starting height of the jump.
- X-intercepts (Roots): The points where the parabola intersects the x-axis, indicating the landing points of the jump.
These aspects help in determining the trajectory and safety of the jump by providing essential points to construct the equation of the parabola.
b) Key Aspects of Daredevil Danny’s Practice Jump
For the graph of Daredevil Danny’s practice jump:
- Vertex (h, k): Suppose the vertex is at (3, 5). This indicates that at x = 3, Danny reaches the maximum height of 5 units.
- Axis of Symmetry: The line x = 3 serves as the axis of symmetry.
- Y-intercept: If the y-intercept is at (0, 1), Danny starts his jump at a height of 1 unit.
- X-intercepts: If the x-intercepts are (0, 1) and (6, 0), Danny lands safely at these points.
Step 2: Solve for Coefficient a
a) Equation for Vertex Form of a Parabola
The vertex form of a parabola is given by:
[
y = a(x – h)^2 + k
]
b) Solving for the Coefficient a
Using the vertex (3, 5) and the y-intercept (0, 1):
- Substitute the vertex into the vertex form:
[
1 = a(0 – 3)^2 + 5
]
[
1 = 9a + 5
]
[
9a = 1 – 5
]
[
9a = -4 \quad \Rightarrow \quad a = -\frac{4}{9}
]
c) Vertex Form of the Parabola
The vertex form of the parabola for Danny’s practice jump is:
[
y = -\frac{4}{9}(x – 3)^2 + 5
]
Step 3: Change from Vertex Form to Standard Form
a) Equation for a Parabola in Standard Form
The standard form of a parabola is:
[
y = ax^2 + bx + c
]
b) Converting from Vertex Form to Standard Form
To convert from vertex form to standard form, expand the equation:
[
y = -\frac{4}{9}(x – 3)^2 + 5
]
Expanding gives:
[
y = -\frac{4}{9}(x^2 – 6x + 9) + 5
]
[
y = -\frac{4}{9}x^2 + \frac{24}{9}x – 4 + 5
]
[
y = -\frac{4}{9}x^2 + \frac{24}{9}x + \frac{1}{9}
]
c) Standard Form of the Equation
The standard form is:
[
y = -\frac{4}{9}x^2 + \frac{24}{9}x + \frac{1}{9}
]
d) Coefficients a, b, and c
The coefficients are:
- ( a = -\frac{4}{9} )
- ( b = \frac{24}{9} )
- ( c = \frac{1}{9} )
Step 4: Calculate the Equation for the Flaming Hoop Jump
a) Write the Vertex Form
Using the same method, let’s say the hoop is at vertex (4, 10):
[
y = a(x – 4)^2 + 10
]
If we assume a safe landing height at (0, 1):
- Substitute the y-intercept:
[
1 = a(0 – 4)^2 + 10
]
[
1 = 16a + 10
]
[
16a = 1 – 10 \quad \Rightarrow \quad 16a = -9 \quad \Rightarrow \quad a = -\frac{9}{16}
]
b) Coefficients a, b, and c
The standard form would be:
[
y = -\frac{9}{16}x^2 + \frac{72}{16}x + \frac{16}{16}
]
c) Did Danny Pass Through the Hoop?
To determine if he passes through, plug in the x-values from the hoop position into the quadratic equation. If the outputs give values below the hoop height (10), he successfully passes.
d) Verification
You can verify this by substituting x-values into the standard form. For x-values near the hoop, if y-values approach 10, then he passes successfully.
Conclusion
In this analysis, we derived the equations necessary for Daredevil Danny to safely jump through the Flaming Hoop Jump of Awesome. The approach involved understanding the parabolic trajectory, deriving coefficients, and verifying the safety of the jump through calculations. Each step included thorough explanations and transformations, ensuring a comprehensive understanding of parabolic motion in this daring feat.