The Graph Of The Function $f(x)=(x+2)(x+6$\] Is Shown Below.Which Statement About The Function Is True?A. The Function Is Positive For All Real Values Of $x$ Where $x \ \textgreater \ -4$.B. The Function Is Negative For All

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Introduction

When analyzing a function, understanding its behavior and characteristics is crucial. In this case, we are given the function $f(x)=(x+2)(x+6)$ and its graph. We need to determine which statement about the function is true. The two statements provided are:

A. The function is positive for all real values of $x$ where $x \ \textgreater \ -4$. B. The function is negative for all real values of $x$ where $x \ \textless \ -6$.

To determine the truth of these statements, we need to analyze the graph of the function and understand its behavior.

Understanding the Graph of the Function

The graph of the function $f(x)=(x+2)(x+6)$ is a quadratic function, which means it has a parabolic shape. The graph is shown below:

Graph of the Function

From the graph, we can see that the function has two x-intercepts at $x=-2$ and $x=-6$. These x-intercepts are the points where the function crosses the x-axis, and they are also the roots of the function.

Analyzing Statement A

Statement A claims that the function is positive for all real values of $x$ where $x \ \textgreater \ -4$. To determine the truth of this statement, we need to analyze the graph of the function in the region where $x \ \textgreater \ -4$.

Analyzing the Graph in the Region $x \ \textgreater \ -4$

From the graph, we can see that the function is above the x-axis in the region where $x \ \textgreater \ -4$. This means that the function is positive in this region.

However, we need to be careful and consider the entire region where $x \ \textgreater \ -4$. We can see that the function is not always positive in this region. For example, at $x=-3$, the function is negative.

Therefore, statement A is not entirely true. The function is not positive for all real values of $x$ where $x \ \textgreater \ -4$.

Analyzing Statement B

Statement B claims that the function is negative for all real values of $x$ where $x \ \textless \ -6$. To determine the truth of this statement, we need to analyze the graph of the function in the region where $x \ \textless \ -6$.

Analyzing the Graph in the Region $x \ \textless \ -6$

From the graph, we can see that the function is below the x-axis in the region where $x \ \textless \ -6$. This means that the function is negative in this region.

Therefore, statement B is true. The function is negative for all real values of $x$ where $x \ \textless \ -6$.

Conclusion

In conclusion, we have analyzed the graph of the function $f(x)=(x+2)(x+6)$ and determined the truth of the two statements provided. We found that statement A is not entirely true, while statement B is true.

The function is not positive for all real values of $x$ where $x \ \textgreater \ -4$, but it is negative for all real values of $x$ where $x \ \textless \ -6$.

Final Thoughts

Understanding the graph of a function is crucial in determining its behavior and characteristics. By analyzing the graph, we can determine the truth of statements about the function and gain a deeper understanding of its behavior.

In this case, we analyzed the graph of the function $f(x)=(x+2)(x+6)$ and determined the truth of the two statements provided. We found that statement A is not entirely true, while statement B is true.

By understanding the graph of a function, we can gain a deeper understanding of its behavior and characteristics, and make informed decisions about its use in various applications.

References

• [1] Graph of the function $f(x)=(x+2)(x+6)$.
• [2] Algebraic analysis of the function $f(x)=(x+2)(x+6)$.

Additional Resources

• [1] Khan Academy: Graphing Quadratic Functions.
• [2] Mathway: Graphing Quadratic Functions.

Keywords

• Graph of the function $f(x)=(x+2)(x+6)$
• Quadratic function
• X-intercepts
• Roots of the function
• Positive and negative regions of the function
• Algebraic analysis of the function
• Graphing quadratic functions
  

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Introduction

In our previous article, we analyzed the graph of the function $f(x)=(x+2)(x+6)$ and determined the truth of two statements about the function. We found that statement A is not entirely true, while statement B is true.

In this article, we will provide a Q&A section to help clarify any questions or doubts about the graph of the function and its behavior.

Q&A Section

Q1: What is the x-intercept of the function $f(x)=(x+2)(x+6)$?

A1: The x-intercepts of the function are the points where the function crosses the x-axis. In this case, the x-intercepts are at $x=-2$ and $x=-6$.

Q2: What is the significance of the x-intercepts in the graph of the function?

A2: The x-intercepts are significant because they represent the roots of the function. The roots of a function are the values of x that make the function equal to zero.

Q3: Is the function $f(x)=(x+2)(x+6)$ a quadratic function?

A3: Yes, the function $f(x)=(x+2)(x+6)$ is a quadratic function. This means that it has a parabolic shape and can be written in the form $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants.

Q4: What is the vertex of the parabola represented by the function $f(x)=(x+2)(x+6)$?

A4: The vertex of the parabola is the point where the parabola changes direction. In this case, the vertex is at $x=-4$.

Q5: Is the function $f(x)=(x+2)(x+6)$ always positive or always negative?

A5: The function $f(x)=(x+2)(x+6)$ is not always positive or always negative. It is positive for some values of x and negative for other values of x.

Q6: How can we determine the truth of statements about the function $f(x)=(x+2)(x+6)$?

A6: We can determine the truth of statements about the function by analyzing the graph of the function. By looking at the graph, we can see where the function is positive or negative and make informed decisions about its behavior.

Q7: What are some real-world applications of the function $f(x)=(x+2)(x+6)$?

A7: The function $f(x)=(x+2)(x+6)$ has many real-world applications, including modeling population growth, predicting stock prices, and analyzing data.

Q8: How can we use the graph of the function $f(x)=(x+2)(x+6)$ to make predictions about its behavior?

A8: We can use the graph of the function to make predictions about its behavior by analyzing the shape of the graph and identifying patterns. By looking at the graph, we can see where the function is increasing or decreasing and make informed decisions about its behavior.

Conclusion

In conclusion, we have provided a Q&A section to help clarify any questions or doubts about the graph of the function $f(x)=(x+2)(x+6)$ and its behavior. We hope that this article has been helpful in understanding the graph of the function and its applications.

Final Thoughts

Understanding the graph of a function is crucial in determining its behavior and characteristics. By analyzing the graph, we can make informed decisions about its use in various applications.

In this article, we have provided a Q&A section to help clarify any questions or doubts about the graph of the function $f(x)=(x+2)(x+6)$ and its behavior. We hope that this article has been helpful in understanding the graph of the function and its applications.

References

• [1] Graph of the function $f(x)=(x+2)(x+6)$.
• [2] Algebraic analysis of the function $f(x)=(x+2)(x+6)$.

Additional Resources

• [1] Khan Academy: Graphing Quadratic Functions.
• [2] Mathway: Graphing Quadratic Functions.

Keywords

• Graph of the function $f(x)=(x+2)(x+6)$
• Quadratic function
• X-intercepts
• Roots of the function
• Positive and negative regions of the function
• Algebraic analysis of the function
• Graphing quadratic functions
• Real-world applications of the function
• Predictions about the function's behavior