Finding the Relationship Between the Functions

Oct 13, 2025 by windailys.com 142 views

Algebra EOC - Common Core Algebra I Test 1 - Edition 3 Question 40 of 57

Understanding the Problem

The problem presents two functions, f(t) and g(t), which describe the number of bacteria in sample A after t minutes. To solve this problem, we need to understand the relationship between these two functions and how they relate to the number of bacteria in sample A.

Function f(t) and Function g(t)

Function f(t) describes the number of bacteria in sample A after t minutes. This function can be represented as f(t) = at^2 + bt + c, where a, b, and c are constants. Function g(t) also describes the number of bacteria in sample A after t minutes, but it is represented as g(t) = dt^2 + et + f, where d, e, and f are constants.

Analyzing the Functions

To analyze the functions, we need to understand the characteristics of each function. Function f(t) is a quadratic function, which means it has a parabolic shape. This shape indicates that the number of bacteria in sample A increases rapidly at first and then slows down as time passes. Function g(t) is also a quadratic function, but it has a different shape than function f(t). This means that the number of bacteria in sample A increases at a different rate than function f(t).

Finding the Relationship Between the Functions

To find the relationship between the functions, we need to compare the two functions and identify any similarities or differences. We can do this by comparing the coefficients of the two functions. The coefficients of function f(t) are a, b, and c, while the coefficients of function g(t) are d, e, and f. By comparing these coefficients, we can identify any patterns or relationships between the two functions.

Solving the Problem

To solve the problem, we need to find the value of t that makes the two functions equal. This means that we need to set f(t) = g(t) and solve for t. We can do this by using algebraic methods, such as substitution or elimination. Once we have found the value of t, we can use it to determine the number of bacteria in sample A at that time.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Set f(t) = g(t) and simplify the equation.
Use algebraic methods to solve for t.
Substitute the value of t into one of the original functions to find the number of bacteria in sample A at that time.

Example Solution

Let's say we have the following functions:

f(t) = 2t^2 + 3t + 1 g(t) = 4t^2 - 2t + 3

To find the value of t that makes the two functions equal, we can set f(t) = g(t) and simplify the equation:

2t^2 + 3t + 1 = 4t^2 - 2t + 3

Combine like terms:

2t^2 - 4t^2 + 3t + 2t + 1 - 3 = 0

Simplify the equation:

-2t^2 + 5t - 2 = 0

Factor the equation:

-(2t - 1)(t - 2) = 0

Solve for t:

2t - 1 = 0 or t - 2 = 0

t = 1/2 or t = 2

Substitute the value of t into one of the original functions to find the number of bacteria in sample A at that time:

f(1/2) = 2(1/2)^2 + 3(1/2) + 1 f(1/2) = 1/2 + 3/2 + 1 f(1/2) = 5/2

Conclusion

In conclusion, the problem presents two functions, f(t) and g(t), which describe the number of bacteria in sample A after t minutes. To solve this problem, we need to understand the relationship between these two functions and how they relate to the number of bacteria in sample A. By analyzing the functions and finding the relationship between them, we can determine the value of t that makes the two functions equal and use it to find the number of bacteria in sample A at that time.

Key Takeaways

The problem presents two functions, f(t) and g(t), which describe the number of bacteria in sample A after t minutes.
To solve the problem, we need to understand the relationship between the two functions and how they relate to the number of bacteria in sample A.
By analyzing the functions and finding the relationship between them, we can determine the value of t that makes the two functions equal and use it to find the number of bacteria in sample A at that time.
The problem requires the use of algebraic methods, such as substitution or elimination, to solve for t.

Common Core Algebra I Test 1 - Edition 3 Question 40 of 57

Answer Key

The answer to the problem is not provided in the question. However, based on the solution provided, the value of t that makes the two functions equal is t = 1/2 or t = 2.
Algebra EOC - Common Core Algebra I Test 1 - Edition 3 Question 40 of 57: Q&A

Q: What is the relationship between the two functions, f(t) and g(t)?

A: The two functions, f(t) and g(t), describe the number of bacteria in sample A after t minutes. Function f(t) is a quadratic function, while function g(t) is also a quadratic function with a different shape.

Q: How do we find the relationship between the two functions?

A: To find the relationship between the two functions, we need to compare the coefficients of the two functions. The coefficients of function f(t) are a, b, and c, while the coefficients of function g(t) are d, e, and f. By comparing these coefficients, we can identify any patterns or relationships between the two functions.

Q: What is the significance of the value of t that makes the two functions equal?

A: The value of t that makes the two functions equal is significant because it allows us to determine the number of bacteria in sample A at that time. By substituting the value of t into one of the original functions, we can find the number of bacteria in sample A at that time.

Q: How do we solve for t?

A: To solve for t, we need to set f(t) = g(t) and simplify the equation. We can then use algebraic methods, such as substitution or elimination, to solve for t.

Q: What are some common algebraic methods used to solve for t?

A: Some common algebraic methods used to solve for t include:

Substitution: This involves substituting the value of one variable into the other equation to solve for the other variable.
Elimination: This involves adding or subtracting the two equations to eliminate one of the variables.
Factoring: This involves factoring the equation to solve for the variable.

Q: What is the importance of understanding the relationship between the two functions?

A: Understanding the relationship between the two functions is important because it allows us to make predictions about the number of bacteria in sample A at different times. By analyzing the functions and finding the relationship between them, we can determine the value of t that makes the two functions equal and use it to make predictions about the number of bacteria in sample A.

Q: How does this problem relate to real-world applications?

A: This problem relates to real-world applications in the field of biology and medicine. Understanding the growth and decay of bacteria is important in the development of new treatments and medications. By analyzing the functions and finding the relationship between them, we can make predictions about the number of bacteria in sample A and develop new treatments and medications.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Not simplifying the equation properly
Not using the correct algebraic methods to solve for t
Not checking the solution for extraneous solutions

Q: How can we check our solution for extraneous solutions?

A: To check our solution for extraneous solutions, we need to plug the value of t back into one of the original functions to see if it is true. If the value of t is not true, then it is an extraneous solution and we need to discard it.

Q: What is the final answer to the problem?

A: The final answer to the problem is not provided in the question. However, based on the solution provided, the value of t that makes the two functions equal is t = 1/2 or t = 2.