Analyzing F(x) = (x+2)(x+6): True Statement Identification

Understanding the behavior of a function, especially its positive and negative intervals, is a fundamental concept in mathematics. Let's dive deep into analyzing the function f(x) = (x+2)(x+6), explore its graphical representation, and identify the true statement about its characteristics. This detailed exploration will not only help in answering the specific question but also enhance your overall understanding of quadratic functions and their graphs.

Understanding the Function and Its Graph

Before we jump into the answer choices, let's break down the function f(x) = (x+2)(x+6). This is a quadratic function, which means its graph will be a parabola. The factored form of the function gives us valuable information about its roots (or x-intercepts). The roots are the values of x for which f(x) = 0. In this case, the roots are x = -2 and x = -6. These are the points where the parabola intersects the x-axis. Knowing the roots is the first crucial step in understanding the graph's behavior. The graph of this function opens upwards because the coefficient of the x² term (which would be 1 if we expanded the expression) is positive. This means the parabola has a minimum point, and the function will be negative between the roots and positive outside of them. Understanding the relationship between the factored form of a quadratic equation and the x-intercepts is a cornerstone of analyzing quadratic functions. This knowledge allows us to quickly visualize the graph's position relative to the x-axis, which is crucial for determining where the function is positive, negative, or zero. Furthermore, the sign of the leading coefficient (the coefficient of the x² term) tells us whether the parabola opens upwards or downwards, dictating whether the vertex is a minimum or maximum point, respectively. Visualizing the parabola helps us predict the function's behavior across different intervals of x-values. For instance, if the parabola opens upwards and has two distinct real roots, as in this case, we can infer that the function will be negative between the roots and positive elsewhere. This is because the y-values on the parabola are negative in the region between the roots (where the parabola dips below the x-axis) and positive in the regions outside the roots (where the parabola rises above the x-axis).

Analyzing the Options

Now, let's consider the options presented in the question. We need to evaluate each statement in the context of our understanding of the function's graph. Remember, we know the parabola opens upwards, and it intersects the x-axis at x = -6 and x = -2. This means the function will be positive for x values less than -6 and greater than -2, and negative for x values between -6 and -2.

Option A: The function is positive for all real values of x where x > -4.

This statement is not entirely accurate. While the function is indeed positive for x > -2, it's not positive for all x > -4. Specifically, between -4 and -2, the function is negative. Therefore, this option is incorrect because it oversimplifies the function's behavior in the interval between its roots. To elaborate, consider a value within the specified range, say x = -3. Plugging this into the function, we get f(-3) = (-3 + 2)(-3 + 6) = (-1)(3) = -3, which is negative. This single counterexample disproves the statement's claim that the function is positive for all x > -4. A more accurate description would consider the intervals where the function is truly positive, which are x < -6 and x > -2. These intervals represent the regions on the number line where the parabola lies above the x-axis.

Option B: The function is negative for all real values of x where -6 < x < -2.

This statement is correct. As we discussed earlier, between the roots x = -6 and x = -2, the parabola lies below the x-axis, indicating that the function is negative in this interval. This statement aligns perfectly with our understanding of quadratic functions and their graphical representation. To further illustrate, let's consider a value within the interval, such as x = -4. Evaluating the function at this point, we have f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4, which confirms that the function is negative. Similarly, any value chosen between -6 and -2 will yield a negative result. This is a direct consequence of the parabola's shape and its position relative to the x-axis. The roots act as boundaries, dividing the x-axis into intervals where the function's sign remains consistent. In this case, the interval between the roots is characterized by negative function values.

Conclusion

Therefore, the true statement about the function f(x) = (x+2)(x+6) is Option B: The function is negative for all real values of x where -6 < x < -2*. Understanding the relationship between a quadratic function, its roots, and its graphical representation is key to solving these types of problems. Remember to always analyze the function's form, identify the roots, and consider the parabola's orientation to accurately determine its behavior across different intervals. For further exploration of quadratic functions and their properties, you can visit resources like Khan Academy's Quadratic Functions section.