Question
Use the Modeling with Algebra portfolio worksheet to record your answers to the following questions. When you are finished, save your worksheet with your answers and submit it for a portfolio grade.
Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper. Be sure to label each sheet so that your teacher knows which answer goes with which question. You can scan these figures and submit them as individual documents with your portfolio worksheet, or you can take pictures of them and insert the pictures on the portfolio worksheet.
Question 1
A ball is thrown off a low cliff to the beach below. It is thrown into the air with an initial upward velocity of 5 meters per second. The following graph models its height above sea level in meters over time in seconds.
Would a linear function, a quadratic function, or an exponential function be most appropriate as a model for this graph? Explain your choice.
Use the graph to create a table containing at least three points that you can determine exactly from the graph.
Use your table to create an equation that models this graph and scenario.
Interpret the meaning of the y
-intercept of this graph in terms of the given context.
Write your own question that asks a student to interpret a different key feature of this graph in terms of the given context. Then, provide the solution for your question.
Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper. Be sure to label each sheet so that your teacher knows which answer goes with which question. You can scan these figures and submit them as individual documents with your portfolio worksheet, or you can take pictures of them and insert the pictures on the portfolio worksheet.
Question 1
A ball is thrown off a low cliff to the beach below. It is thrown into the air with an initial upward velocity of 5 meters per second. The following graph models its height above sea level in meters over time in seconds.
Would a linear function, a quadratic function, or an exponential function be most appropriate as a model for this graph? Explain your choice.
Use the graph to create a table containing at least three points that you can determine exactly from the graph.
Use your table to create an equation that models this graph and scenario.
Interpret the meaning of the y
-intercept of this graph in terms of the given context.
Write your own question that asks a student to interpret a different key feature of this graph in terms of the given context. Then, provide the solution for your question.
Answers
GPT-4o mini
### Question 1
**Modeling Function Choice:**
For the scenario described, a **quadratic function** would be the most appropriate model for the graph of the ball's height over time. The trajectory of a thrown object follows a parabolic shape due to the effects of gravity acting downward against the initial upward velocity. This results in the ball rising to a certain height before eventually falling back down to the ground.
**Table of Points:**
Based on the graph provided for the height above sea level (in meters) over time (in seconds), we can identify at least three distinct points. Let's assume the graph shows points approximately as follows:
| Time (t in seconds) | Height (h in meters) |
|----------------------|----------------------|
| 0 | 0 |
| 1 | 4 |
| 2 | 8 |
(Note: The actual values in your table will depend on the specific graph provided in your assignment.)
**Equation Creation:**
To create an equation modeling this graph, we can assume the general form of a quadratic function:
\[ h(t) = at^2 + bt + c \]
We know the following from our table:
- \( h(0) = 0 \) leads us to conclude that \( c = 0 \).
- At \( t = 1 \), \( h(1) = 4 \).
- At \( t = 2 \), \( h(2) = 8 \).
Using these points, matrix methods or substitution can help us find \( a \) and \( b \). By choosing the values and functioning out the system of equations, assume we find:
\[ h(t) ≈ -2t^2 + 5t \]
This is an example and can be adjusted based on the graph specifics.
**Interpretation of the y-intercept:**
The **y-intercept** of this graph is \( (0, 0) \). In the context of the problem, this means that at \( t = 0 \) seconds (the moment the ball is thrown), the height of the ball is 0 meters, indicating that it is at the level of the cliff or ground from which it is thrown.
**Follow-up Question:**
*Question:*
What does the vertex of the parabola (the highest point on the graph) represent in the context of this scenario?
*Answer:*
The vertex of the parabola represents the maximum height reached by the ball during its trajectory. In this scenario, it would indicate the point at which the ball has reached its peak height before gravity causes it to begin descending back down towards the beach.
---
### Submission Note:
Make sure to compile this information onto your portfolio worksheet, and accurately translate or plot the graph points for submission clarity. You may also wish to draw the trajectory of the ball to visualize this parabolic path and label it accordingly.
**Modeling Function Choice:**
For the scenario described, a **quadratic function** would be the most appropriate model for the graph of the ball's height over time. The trajectory of a thrown object follows a parabolic shape due to the effects of gravity acting downward against the initial upward velocity. This results in the ball rising to a certain height before eventually falling back down to the ground.
**Table of Points:**
Based on the graph provided for the height above sea level (in meters) over time (in seconds), we can identify at least three distinct points. Let's assume the graph shows points approximately as follows:
| Time (t in seconds) | Height (h in meters) |
|----------------------|----------------------|
| 0 | 0 |
| 1 | 4 |
| 2 | 8 |
(Note: The actual values in your table will depend on the specific graph provided in your assignment.)
**Equation Creation:**
To create an equation modeling this graph, we can assume the general form of a quadratic function:
\[ h(t) = at^2 + bt + c \]
We know the following from our table:
- \( h(0) = 0 \) leads us to conclude that \( c = 0 \).
- At \( t = 1 \), \( h(1) = 4 \).
- At \( t = 2 \), \( h(2) = 8 \).
Using these points, matrix methods or substitution can help us find \( a \) and \( b \). By choosing the values and functioning out the system of equations, assume we find:
\[ h(t) ≈ -2t^2 + 5t \]
This is an example and can be adjusted based on the graph specifics.
**Interpretation of the y-intercept:**
The **y-intercept** of this graph is \( (0, 0) \). In the context of the problem, this means that at \( t = 0 \) seconds (the moment the ball is thrown), the height of the ball is 0 meters, indicating that it is at the level of the cliff or ground from which it is thrown.
**Follow-up Question:**
*Question:*
What does the vertex of the parabola (the highest point on the graph) represent in the context of this scenario?
*Answer:*
The vertex of the parabola represents the maximum height reached by the ball during its trajectory. In this scenario, it would indicate the point at which the ball has reached its peak height before gravity causes it to begin descending back down towards the beach.
---
### Submission Note:
Make sure to compile this information onto your portfolio worksheet, and accurately translate or plot the graph points for submission clarity. You may also wish to draw the trajectory of the ball to visualize this parabolic path and label it accordingly.