The Graph Of The Function $f(x)=-(x+6)(x+2)$ Is Shown Below.Which Statement About The Function Is True?A. The Function Is Increasing For All Real Values Of $x$ Where \$x \ \textless \ -4$[/tex\].B. The Function Is
Introduction
When analyzing a function, it's essential to understand its behavior, including its increasing and decreasing intervals. The graph of the function $f(x)=-(x+6)(x+2)$ provides valuable insights into the function's behavior. In this article, we will examine the graph and determine which statement about the function is true.
Understanding the Function
The given function is a quadratic function in the form of $f(x)=a(x-h)(x-k)$. In this case, $a=-1$, $h=-6$, and $k=-2$. The graph of a quadratic function is a parabola that opens upward or downward, depending on the sign of $a$. Since $a=-1$, the parabola opens downward.
Graph Analysis
The graph of the function $f(x)=-(x+6)(x+2)$ is shown below. The graph is a downward-opening parabola with its vertex at $(-4, 0)$. The parabola intersects the x-axis at $x=-6$ and $x=-2$.
Increasing and Decreasing Intervals
To determine the increasing and decreasing intervals of the function, we need to examine the sign of the function's derivative. The derivative of the function is given by $f'(x)=-2(x+4)$. The derivative is zero when $x=-4$, which is the x-coordinate of the vertex.
Analysis of Statement A
Statement A claims that the function is increasing for all real values of $x$ where $x<-4$. To verify this statement, we need to examine the sign of the derivative in the interval $x<-4$. Since the derivative is negative in this interval, the function is decreasing, not increasing.
Analysis of Statement B
Statement B claims that the function is decreasing for all real values of $x$ where $x>-4$. To verify this statement, we need to examine the sign of the derivative in the interval $x>-4$. Since the derivative is positive in this interval, the function is increasing, not decreasing.
Conclusion
Based on the analysis of the graph and the derivative of the function, we can conclude that the function $f(x)=-(x+6)(x+2)$ is decreasing for all real values of $x$ where $x<-4$ and increasing for all real values of $x$ where $x>-4$. Therefore, statement A is false, and statement B is also false.
Final Answer
The final answer is that the function $f(x)=-(x+6)(x+2)$ is decreasing for all real values of $x$ where $x<-4$ and increasing for all real values of $x$ where $x>-4$.
Additional Information
- The vertex of the parabola is at $(-4, 0)$.
- The parabola intersects the x-axis at $x=-6$ and $x=-2$.
- The derivative of the function is $f'(x)=-2(x+4)$.
References
Graph of the Function
The graph of the function $f(x)=-(x+6)(x+2)$ is a downward-opening parabola with its vertex at $(-4, 0)$. The parabola intersects the x-axis at $x=-6$ and $x=-2$.
Derivative of the Function
The derivative of the function is given by $f'(x)=-2(x+4)$. The derivative is zero when $x=-4$, which is the x-coordinate of the vertex.
Increasing and Decreasing Intervals
To determine the increasing and decreasing intervals of the function, we need to examine the sign of the derivative. The derivative is negative in the interval $x<-4$, which means the function is decreasing in this interval. The derivative is positive in the interval $x>-4$, which means the function is increasing in this interval.
Final Answer
The final answer is that the function $f(x)=-(x+6)(x+2)$ is decreasing for all real values of $x$ where $x<-4$ and increasing for all real values of $x$ where $x>-4$.
Introduction
In our previous article, we analyzed the graph of the function $f(x)=-(x+6)(x+2)$ and determined that the function is decreasing for all real values of $x$ where $x<-4$ and increasing for all real values of $x$ where $x>-4$. In this article, we will answer some frequently asked questions about the graph of the function.
Q1: What is the vertex of the parabola?
A1: The vertex of the parabola is at $(-4, 0)$.
Q2: Where does the parabola intersect the x-axis?
A2: The parabola intersects the x-axis at $x=-6$ and $x=-2$.
Q3: What is the derivative of the function?
A3: The derivative of the function is $f'(x)=-2(x+4)$.
Q4: Is the function increasing or decreasing in the interval $x<-4$?
A4: The function is decreasing in the interval $x<-4$.
Q5: Is the function increasing or decreasing in the interval $x>-4$?
A5: The function is increasing in the interval $x>-4$.
Q6: What is the x-coordinate of the vertex?
A6: The x-coordinate of the vertex is $-4$.
Q7: What is the y-coordinate of the vertex?
A7: The y-coordinate of the vertex is $0$.
Q8: Is the parabola a downward-opening or upward-opening parabola?
A8: The parabola is a downward-opening parabola.
Q9: What is the sign of the derivative in the interval $x<-4$?
A9: The derivative is negative in the interval $x<-4$.
Q10: What is the sign of the derivative in the interval $x>-4$?
A10: The derivative is positive in the interval $x>-4$.
Conclusion
In this article, we answered some frequently asked questions about the graph of the function $f(x)=-(x+6)(x+2)$. We hope that this article has provided you with a better understanding of the graph of the function.
Additional Information
Graph of the Function
The graph of the function $f(x)=-(x+6)(x+2)$ is a downward-opening parabola with its vertex at $(-4, 0)$. The parabola intersects the x-axis at $x=-6$ and $x=-2$.
Derivative of the Function
The derivative of the function is given by $f'(x)=-2(x+4)$. The derivative is zero when $x=-4$, which is the x-coordinate of the vertex.
Increasing and Decreasing Intervals
To determine the increasing and decreasing intervals of the function, we need to examine the sign of the derivative. The derivative is negative in the interval $x<-4$, which means the function is decreasing in this interval. The derivative is positive in the interval $x>-4$, which means the function is increasing in this interval.
Final Answer
The final answer is that the function $f(x)=-(x+6)(x+2)$ is decreasing for all real values of $x$ where $x<-4$ and increasing for all real values of $x$ where $x>-4$.