Zander Was Given Two Functions: One Represented By The Graph And The Function F ( X ) = ( X + 4 ) 2 F(x)=(x+4)^2 F ( X ) = ( X + 4 ) 2 . What Can He Conclude About The Two Functions?A. They Have The Same Vertex. B. They Have One X X X -intercept That Is The Same. C. They Have

Introduction

In mathematics, functions are used to describe relationships between variables. When comparing two functions, we often look for similarities and differences in their behavior, such as their domain, range, and key points like the vertex or x-intercepts. In this article, we will explore the function $f(x)=(x+4)^2$ and compare it to another function represented by a graph. We will examine the properties of this function and determine what conclusions can be drawn about the two functions.

Understanding the Function $f(x)=(x+4)^2$

The function $f(x)=(x+4)^2$ is a quadratic function, which means it can be written in the form $f(x)=ax^2+bx+c$. In this case, the function is in the form of a perfect square trinomial, where $a=1$, $b=8$, and $c=16$. This function has a vertex at $(-4, 0)$, which is the minimum point of the parabola.

Key Properties of the Function

• Vertex: The vertex of the function is at $(-4, 0)$.
• Axis of Symmetry: The axis of symmetry is the vertical line $x=-4$.
• Domain and Range: The domain of the function is all real numbers, and the range is all non-negative real numbers.
• X-Intercepts: The x-intercepts of the function are at $(-6, 0)$ and $(-2, 0)$.

Comparing the Function to Another Graph

Let's assume that the graph of another function is given, and we want to compare it to the function $f(x)=(x+4)^2$. We will examine the properties of the other function and determine what conclusions can be drawn about the two functions.

Conclusion 1: Same Vertex

If the other function has the same vertex as the function $f(x)=(x+4)^2$, then we can conclude that the two functions have the same vertex. This means that the minimum point of the parabola is the same for both functions.

Conclusion 2: Same X-Intercept

If the other function has the same x-intercept as the function $f(x)=(x+4)^2$, then we can conclude that the two functions have one x-intercept that is the same. This means that the point where the graph of the function crosses the x-axis is the same for both functions.

Conclusion 3: Different Functions

If the other function does not have the same vertex or x-intercept as the function $f(x)=(x+4)^2$, then we can conclude that the two functions are different. This means that the behavior of the two functions is different, and they do not have the same key points.

Conclusion

In conclusion, when comparing two functions, we can draw conclusions about their properties, such as their vertex, x-intercepts, and axis of symmetry. By examining the properties of the function $f(x)=(x+4)^2$, we can determine what conclusions can be drawn about the two functions. If the other function has the same vertex or x-intercept, then we can conclude that the two functions have the same vertex or one x-intercept that is the same. If the other function does not have the same vertex or x-intercept, then we can conclude that the two functions are different.

Key Takeaways

• The function $f(x)=(x+4)^2$ is a quadratic function with a vertex at $(-4, 0)$.
• The function has a domain of all real numbers and a range of all non-negative real numbers.
• The x-intercepts of the function are at $(-6, 0)$ and $(-2, 0)$.
• When comparing two functions, we can draw conclusions about their properties, such as their vertex, x-intercepts, and axis of symmetry.

Final Thoughts

Q: What is the vertex of the function $f(x)=(x+4)^2$?

A: The vertex of the function $f(x)=(x+4)^2$ is at $(-4, 0)$.

Q: What is the axis of symmetry of the function $f(x)=(x+4)^2$?

A: The axis of symmetry of the function $f(x)=(x+4)^2$ is the vertical line $x=-4$.

Q: What is the domain and range of the function $f(x)=(x+4)^2$?

A: The domain of the function $f(x)=(x+4)^2$ is all real numbers, and the range is all non-negative real numbers.

Q: How many x-intercepts does the function $f(x)=(x+4)^2$ have?

A: The function $f(x)=(x+4)^2$ has two x-intercepts at $(-6, 0)$ and $(-2, 0)$.

Q: What can be concluded if the other function has the same vertex as the function $f(x)=(x+4)^2$?

A: If the other function has the same vertex as the function $f(x)=(x+4)^2$, then we can conclude that the two functions have the same vertex.

Q: What can be concluded if the other function has the same x-intercept as the function $f(x)=(x+4)^2$?

A: If the other function has the same x-intercept as the function $f(x)=(x+4)^2$, then we can conclude that the two functions have one x-intercept that is the same.

Q: What can be concluded if the other function does not have the same vertex or x-intercept as the function $f(x)=(x+4)^2$?

A: If the other function does not have the same vertex or x-intercept as the function $f(x)=(x+4)^2$, then we can conclude that the two functions are different.

Q: How can we compare two functions?

A: We can compare two functions by examining their properties, such as their vertex, x-intercepts, and axis of symmetry.

Q: What are some key takeaways from comparing functions?

A: Some key takeaways from comparing functions include:

• The function $f(x)=(x+4)^2$ is a quadratic function with a vertex at $(-4, 0)$.
• The function has a domain of all real numbers and a range of all non-negative real numbers.
• The x-intercepts of the function are at $(-6, 0)$ and $(-2, 0)$.
• When comparing two functions, we can draw conclusions about their properties, such as their vertex, x-intercepts, and axis of symmetry.

Q: What is the final thought on comparing functions?

A: In conclusion, the function $f(x)=(x+4)^2$ is a quadratic function with a vertex at $(-4, 0)$. When comparing this function to another graph, we can draw conclusions about their properties, such as their vertex, x-intercepts, and axis of symmetry. By examining the properties of the function, we can determine what conclusions can be drawn about the two functions.