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 Functions?- The Vertex Of

Introduction

In mathematics, functions are used to describe the relationship between variables. Quadratic functions, in particular, are used to model various real-world phenomena, such as the trajectory of a projectile or the growth of a population. In this article, we will explore the relationship between two quadratic functions, $f(x)$ and $g(x)$, and determine which statements are true about their relationship.

The Functions

The two quadratic functions are given by:

\begin{align*} f(x) &= 4(x-3)^2 + 6 \ g(x) &= -2(x+1)^2 + 4 \end{align*}

Vertex Form

To understand the relationship between the two functions, we need to express them in vertex form. The vertex form of a quadratic function is given by:

$$f(x) = a(x-h)^2 + k$$

where $(h,k)$ is the vertex of the parabola.

Vertex of $f(x)$

The vertex of $f(x)$ can be found by completing the square:

\begin{align*} f(x) &= 4(x-3)^2 + 6 \ &= 4(x^2 - 6x + 9) + 6 \ &= 4x^2 - 24x + 36 + 6 \ &= 4x^2 - 24x + 42 \end{align*}

The vertex of $f(x)$ is $(3, 42)$.

Vertex of $g(x)$

The vertex of $g(x)$ can be found by completing the square:

\begin{align*} g(x) &= -2(x+1)^2 + 4 \ &= -2(x^2 + 2x + 1) + 4 \ &= -2x^2 - 4x - 2 + 4 \ &= -2x^2 - 4x + 2 \end{align*}

The vertex of $g(x)$ is $(-1, 2)$.

Relationship Between the Functions

Now that we have expressed the functions in vertex form, we can compare their vertices. The vertex of $f(x)$ is $(3, 42)$, while the vertex of $g(x)$ is $(-1, 2)$. We can see that the two functions have different vertices, which means that they have different shapes.

Statements About the Relationship

Based on the information above, we can make the following statements about the relationship between the functions:

• The functions have different vertices: This is true, as we have shown that the vertices of $f(x)$ and $g(x)$ are different.
• The functions have different shapes: This is true, as the functions have different vertices, which means that they have different shapes.
• The functions are not congruent: This is true, as the functions have different vertices and shapes, which means that they are not congruent.
• The functions are not similar: This is true, as the functions have different vertices and shapes, which means that they are not similar.

Conclusion

In conclusion, we have explored the relationship between two quadratic functions, $f(x)$ and $g(x)$. We have expressed the functions in vertex form and compared their vertices. Based on the information above, we have made several statements about the relationship between the functions. These statements are:

• The functions have different vertices.
• The functions have different shapes.
• The functions are not congruent.
• The functions are not similar.

Final Answer

The final answer is:

• The functions have different vertices.
• The functions have different shapes.
• The functions are not congruent.
• The functions are not similar.

Introduction

In our previous article, we explored the relationship between two quadratic functions, $f(x)$ and $g(x)$. We expressed the functions in vertex form and compared their vertices. In this article, we will answer some frequently asked questions about the relationship between the functions.

Q: What is the difference between the two functions?

A: The two functions have different vertices, which means that they have different shapes. The vertex of $f(x)$ is $(3, 42)$, while the vertex of $g(x)$ is $(-1, 2)$. This difference in vertices results in different shapes for the two functions.

Q: Are the functions congruent?

A: No, the functions are not congruent. Congruent functions have the same shape and size, but the two functions have different vertices and shapes, which means that they are not congruent.

Q: Are the functions similar?

A: No, the functions are not similar. Similar functions have the same shape but different sizes. The two functions have different vertices and shapes, which means that they are not similar.

Q: How can I determine if two functions are congruent or similar?

A: To determine if two functions are congruent or similar, you need to compare their vertices and shapes. If the vertices and shapes are the same, then the functions are congruent. If the vertices and shapes are the same but the sizes are different, then the functions are similar.

Q: Can I use the vertex form to compare the functions?

A: Yes, you can use the vertex form to compare the functions. The vertex form of a quadratic function is given by:

$$f(x) = a(x-h)^2 + k$$

where $(h,k)$ is the vertex of the parabola. By comparing the vertices of the two functions, you can determine if they are congruent or similar.

Q: What is the significance of the vertex form in understanding the relationship between the functions?

A: The vertex form is significant in understanding the relationship between the functions because it allows you to compare the vertices and shapes of the functions. By comparing the vertices and shapes, you can determine if the functions are congruent or similar.

Q: Can I use other methods to compare the functions?

A: Yes, you can use other methods to compare the functions. For example, you can use the graphing method to compare the functions. By graphing the functions, you can visually compare their shapes and determine if they are congruent or similar.

Conclusion

In conclusion, we have answered some frequently asked questions about the relationship between two quadratic functions, $f(x)$ and $g(x)$. We have discussed the difference between the functions, whether they are congruent or similar, and how to determine if two functions are congruent or similar. We have also discussed the significance of the vertex form in understanding the relationship between the functions.

Final Answer

The final answer is:

• The functions have different vertices and shapes.
• The functions are not congruent.
• The functions are not similar.
• The vertex form is significant in understanding the relationship between the functions.
• Other methods, such as graphing, can be used to compare the functions.