Select All The Correct Answers.Consider Functions { F $}$ And { G $} . . . { \begin{align*} f(x) &= 4(x-3)^2 + 6 \\ g(x) &= -2(x+1)^2 + 4 \end{align*} \} Which Statements Are True About The Relationship Between The

by ADMIN 215 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When dealing with two functions, it's essential to understand how they relate to each other. In this article, we will explore the relationship between two given functions, f(x) and g(x), and determine which statements are true about their relationship.

The Functions

The two functions given are:

  • f(x) = 4(x-3)^2 + 6
  • g(x) = -2(x+1)^2 + 4

Analyzing the Functions

To understand the relationship between the two functions, we need to analyze their properties. Let's start by examining the general form of a quadratic function, which is:

ax^2 + bx + c

In this form, 'a' is the coefficient of the squared term, 'b' is the coefficient of the linear term, and 'c' is the constant term.

Comparing the Functions

Now, let's compare the two functions:

  • f(x) = 4(x-3)^2 + 6
  • g(x) = -2(x+1)^2 + 4

We can see that both functions are quadratic, but they have different coefficients and constant terms.

Determining the Relationship

To determine the relationship between the two functions, we need to examine their graphs. Let's start by graphing both functions:

Graphing the Functions

The graph of f(x) is a parabola that opens upwards, with a vertex at (3, 6). The graph of g(x) is a parabola that opens downwards, with a vertex at (-1, 4).

Comparing the Graphs

Now, let's compare the graphs of the two functions:

  • f(x)
  • g(x)

We can see that the graphs of the two functions are reflections of each other across the x-axis.

Determining the Correct Statements

Based on our analysis, we can determine which statements are true about the relationship between the two functions.

Statement 1: The graphs of f(x) and g(x) are reflections of each other across the x-axis.

This statement is true. The graphs of the two functions are reflections of each other across the x-axis.

Statement 2: The functions f(x) and g(x) are inverses of each other.

This statement is false. The functions f(x) and g(x) are not inverses of each other.

Statement 3: The functions f(x) and g(x) have the same vertex.

This statement is false. The functions f(x) and g(x) have different vertices.

Statement 4: The functions f(x) and g(x) have the same axis of symmetry.

This statement is false. The functions f(x) and g(x) have different axes of symmetry.

Statement 5: The functions f(x) and g(x) are reflections of each other across the y-axis.

This statement is false. The functions f(x) and g(x) are reflections of each other across the x-axis.

Statement 6: The functions f(x) and g(x) have the same leading coefficient.

This statement is false. The functions f(x) and g(x) have different leading coefficients.

Statement 7: The functions f(x) and g(x) have the same constant term.

This statement is false. The functions f(x) and g(x) have different constant terms.

Statement 8: The functions f(x) and g(x) are reflections of each other across the line y = x.

This statement is false. The functions f(x) and g(x) are reflections of each other across the x-axis.

Statement 9: The functions f(x) and g(x) have the same axis of symmetry.

This statement is false. The functions f(x) and g(x) have different axes of symmetry.

Statement 10: The functions f(x) and g(x) are reflections of each other across the line y = -x.

This statement is false. The functions f(x) and g(x) are reflections of each other across the x-axis.

Conclusion

In conclusion, the correct statements about the relationship between the two functions are:

  • The graphs of f(x) and g(x) are reflections of each other across the x-axis.

Introduction

In our previous article, we explored the relationship between two given functions, f(x) and g(x). We analyzed their properties, compared their graphs, and determined which statements are true about their relationship. In this article, we will answer some frequently asked questions about the relationship between two functions.

Q: What is the relationship between the two functions?

A: The two functions, f(x) and g(x), are reflections of each other across the x-axis.

Q: How can we determine the relationship between two functions?

A: To determine the relationship between two functions, we need to analyze their properties, compare their graphs, and examine their equations.

Q: What are some common relationships between functions?

A: Some common relationships between functions include:

  • Reflections: One function is a reflection of the other across the x-axis, y-axis, or a line.
  • Translations: One function is a translation of the other by a certain distance.
  • Rotations: One function is a rotation of the other by a certain angle.
  • Dilations: One function is a dilation of the other by a certain scale factor.

Q: How can we graph two functions on the same coordinate plane?

A: To graph two functions on the same coordinate plane, we need to:

  • Plot the graphs of each function separately.
  • Use a different color or line style for each function.
  • Label each graph with its corresponding equation.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are reflections of each other across the line y = x. The inverse of a function is obtained by swapping the x and y variables in the equation.

Q: How can we find the inverse of a function?

A: To find the inverse of a function, we need to:

  • Swap the x and y variables in the equation.
  • Solve for y.
  • Write the resulting equation in terms of x.

Q: What is the difference between a function and its reciprocal?

A: A function and its reciprocal are two functions that are reflections of each other across the x-axis. The reciprocal of a function is obtained by taking the reciprocal of the function's output.

Q: How can we find the reciprocal of a function?

A: To find the reciprocal of a function, we need to:

  • Take the reciprocal of the function's output.
  • Write the resulting equation in terms of x.

Q: What is the difference between a function and its absolute value?

A: A function and its absolute value are two functions that are reflections of each other across the x-axis. The absolute value of a function is obtained by taking the absolute value of the function's output.

Q: How can we find the absolute value of a function?

A: To find the absolute value of a function, we need to:

  • Take the absolute value of the function's output.
  • Write the resulting equation in terms of x.

Conclusion

In conclusion, understanding the relationship between two functions is an essential concept in mathematics. By analyzing their properties, comparing their graphs, and examining their equations, we can determine the relationship between two functions. We hope this Q&A article has provided you with a better understanding of the relationship between two functions.

Additional Resources

  • Graphing Functions: A tutorial on graphing functions.
  • Function Properties: A tutorial on function properties.
  • Inverse Functions: A tutorial on inverse functions.
  • Reciprocal Functions: A tutorial on reciprocal functions.
  • Absolute Value Functions: A tutorial on absolute value functions.