The Graph Of The Function F ( X ) = ( X + 2 ) ( X − 4 F(x)=(x+2)(x-4 F ( X ) = ( X + 2 ) ( X − 4 ] Is Shown.Which Describes All Of The Values For Which The Graph Is Negative And Increasing?A. All Real Values Of X X X Where X \textless − 2 X \ \textless \ -2 X \textless − 2 B. All Real Values Of X X X

Introduction

When analyzing the graph of a function, it's essential to understand the behavior of the function in different intervals. In this case, we're given the function $f(x)=(x+2)(x-4)$ and asked to describe all the values for which the graph is negative and increasing. To tackle this problem, we need to understand the properties of the function, including its roots, intervals of increase and decrease, and the behavior of the function in different intervals.

Understanding the Function

The given function is a quadratic function in the form of $f(x)=ax^2+bx+c$. In this case, the function can be rewritten as $f(x)=x^2-2x-8$. The roots of the function can be found by setting the function equal to zero and solving for $x$. In this case, the roots are $x=-2$ and $x=4$.

Identifying Intervals of Increase and Decrease

To determine the intervals of increase and decrease, we need to find the critical points of the function. The critical points occur when the derivative of the function is equal to zero or undefined. In this case, the derivative of the function is $f'(x)=2x-2$. Setting the derivative equal to zero, we get $2x-2=0$, which gives us $x=1$. This is the only critical point of the function.

Analyzing the Graph

Now that we have identified the roots and critical points of the function, we can analyze the graph. The graph of the function is a parabola that opens upward. The roots of the function are $x=-2$ and $x=4$, which are the points where the graph intersects the x-axis. The critical point of the function is $x=1$, which is the point where the graph changes from decreasing to increasing.

Describing Negative and Increasing Values

To describe all the values for which the graph is negative and increasing, we need to find the intervals where the function is negative and the derivative is positive. The function is negative when $x<-2$ or $x>4$. The derivative is positive when $x>1$. Therefore, the graph is negative and increasing when $x>1$ and $x>4$.

Conclusion

In conclusion, the graph of the function $f(x)=(x+2)(x-4)$ is negative and increasing when $x>1$ and $x>4$. This means that the graph is negative and increasing for all real values of $x$ greater than 1 and 4.

Final Answer

The final answer is A. All real values of $x$ where $x \ \textless \ -2$

Introduction

In our previous article, we explored the graph of the function $f(x)=(x+2)(x-4)$ and described all the values for which the graph is negative and increasing. In this article, we'll answer some frequently asked questions related to the graph of the function and provide additional insights into the behavior of the function.

Q&A

Q: What are the roots of the function $f(x)=(x+2)(x-4)$?

A: The roots of the function are $x=-2$ and $x=4$. These are the points where the graph intersects the x-axis.

Q: What is the critical point of the function $f(x)=(x+2)(x-4)$?

A: The critical point of the function is $x=1$. This is the point where the graph changes from decreasing to increasing.

Q: When is the function $f(x)=(x+2)(x-4)$ negative?

A: The function is negative when $x<-2$ or $x>4$.

Q: When is the derivative of the function $f(x)=(x+2)(x-4)$ positive?

A: The derivative is positive when $x>1$.

Q: When is the graph of the function $f(x)=(x+2)(x-4)$ negative and increasing?

A: The graph is negative and increasing when $x>1$ and $x>4$.

Q: What is the final answer to the problem?

A: The final answer is A. All real values of $x$ where $x \ \textless \ -2$.

Q: Can you provide a visual representation of the graph of the function $f(x)=(x+2)(x-4)$?

A: Yes, here is a visual representation of the graph:

# Graph of the Function f(x) = (x+2)(x-4)
x-axis
-2 | 0 | 1 | 2 | 3 | 4 | 5
y-axis
-10 | -8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 | 10

Q: How can I use the graph of the function $f(x)=(x+2)(x-4)$ to solve problems?

A: You can use the graph to identify the intervals where the function is negative, increasing, or decreasing. You can also use the graph to find the roots and critical points of the function.

Conclusion

In conclusion, the graph of the function $f(x)=(x+2)(x-4)$ is a powerful tool for understanding the behavior of the function. By analyzing the graph, we can identify the intervals where the function is negative, increasing, or decreasing, and use this information to solve problems.

Final Thoughts

The graph of the function $f(x)=(x+2)(x-4)$ is a classic example of a quadratic function. By understanding the properties of the function, including its roots, critical points, and intervals of increase and decrease, we can use the graph to solve a wide range of problems. Whether you're a student, teacher, or professional, the graph of the function $f(x)=(x+2)(x-4)$ is an essential tool for anyone working with quadratic functions.