Analyzing The Graph Of F(x) = -(x+3)(x-1) Determining True Statements

Understanding the behavior of functions is a cornerstone of mathematics, and the function f(x) = -(x+3)(x-1) offers a compelling case study. In this comprehensive analysis, we will delve into the intricacies of this quadratic function, exploring its graph, key features, and the truth behind various statements about its nature. Our exploration will not only solidify your understanding of this specific function but also equip you with the tools to analyze similar functions in the future. This analysis aims to provide a clear and insightful explanation, perfect for students, educators, and anyone with a passion for mathematics.

Dissecting the Function f(x) = -(x+3)(x-1)

The provided function, f(x) = -(x+3)(x-1), is a quadratic function presented in its factored form. This form provides immediate insights into the function's roots, which are the values of x for which f(x) = 0. By setting each factor to zero, we find the roots to be x = -3 and x = 1. These roots represent the points where the graph of the function intersects the x-axis. Understanding the roots is crucial for sketching the graph and analyzing the function's behavior.

Beyond the roots, the factored form also reveals the function's overall shape. The negative sign in front of the expression indicates that the parabola opens downwards. This means the function has a maximum value rather than a minimum value. The coefficient of the x² term, which can be found by expanding the expression, determines the parabola's width. In this case, expanding -(x+3)(x-1) gives us -x² - 2x + 3, indicating a coefficient of -1 for the x² term. This information helps us visualize the parabola's steepness.

To further understand the function, we can determine its vertex, which is the point where the function reaches its maximum value. The x-coordinate of the vertex lies exactly halfway between the roots. In this case, the x-coordinate of the vertex is (-3 + 1)/2 = -1. To find the y-coordinate of the vertex, we substitute this x-value back into the function: f(-1) = -(-1+3)(-1-1) = - (2)(-2) = 4. Therefore, the vertex of the parabola is at the point (-1, 4). The vertex is a crucial point for understanding the function's range and its overall behavior.

Finally, analyzing the leading coefficient and the roots allows us to determine the intervals where the function is positive and negative. Since the parabola opens downwards, the function will be positive between the roots and negative outside of them. This understanding is fundamental to evaluating the statements about the function's behavior in different intervals.

Evaluating Statement A: The Function is Positive for All Real Values of x Where x < -1

Statement A asserts that the function f(x) = -(x+3)(x-1) is positive for all real values of x where x < -1. To rigorously evaluate this statement, we need to consider the graph of the function and its roots. As we established earlier, the roots of the function are x = -3 and x = 1. These roots divide the number line into three intervals: x < -3, -3 < x < 1, and x > 1. The behavior of the function in each of these intervals determines the truth of the statement.

Within the interval x < -3, both factors (x+3) and (x-1) are negative. Therefore, their product is positive. However, the negative sign in front of the expression -(x+3)(x-1) makes the overall function value negative in this interval. For example, if we take x = -4, we get f(-4) = -(-4+3)(-4-1) = -(-1)(-5) = -5, which is clearly negative. This directly contradicts the claim that the function is positive for all x < -1.

In the interval -3 < x < 1, the factor (x+3) is positive, and the factor (x-1) is negative. Their product is therefore negative. However, the negative sign in front of the expression makes the overall function value positive in this interval. This is the region between the roots where the parabola lies above the x-axis. For instance, if we take x = 0, we get f(0) = -(0+3)(0-1) = -(3)(-1) = 3, which is positive.

For the interval x > 1, both factors (x+3) and (x-1) are positive, making their product positive. The negative sign in front of the expression then makes the overall function value negative. For example, if we take x = 2, we get f(2) = -(2+3)(2-1) = -(5)(1) = -5, which is negative.

Based on this analysis, it is evident that Statement A is false. The function is not positive for all real values of x where x < -1. Instead, it is negative in the interval x < -3. The function is positive only in the interval between the roots, -3 < x < 1. Therefore, a clear understanding of the roots and the parabola's orientation is crucial for accurately determining the function's behavior.

Evaluating Statement B: The Function is Negative for All Real Values of x Where x < -3 and Where x > 1

Statement B proposes that the function f(x) = -(x+3)(x-1) is negative for all real values of x where x < -3 and where x > 1. To assess the validity of this statement, we again turn to our understanding of the function's graph and its roots. As we established, the roots of the function are x = -3 and x = 1, and the parabola opens downwards due to the negative sign in front of the expression.

For the interval x < -3, both factors (x+3) and (x-1) are negative. The product of two negative numbers is positive. However, the negative sign preceding the factored expression makes the entire function negative. To illustrate, let's consider x = -4. We find f(-4) = -(-4+3)(-4-1) = -(-1)(-5) = -5, which is indeed negative. This supports the statement's claim for x < -3.

Considering the interval -3 < x < 1, we know that (x+3) is positive and (x-1) is negative. The product of a positive and a negative number is negative, but the negative sign in front of the factored expression makes the entire function positive in this interval. This is consistent with the parabola lying above the x-axis between its roots. For example, at x = 0, we have f(0) = -(0+3)(0-1) = -(3)(-1) = 3, which is positive. This interval is critical in distinguishing the regions where the function is positive versus negative.

Now, let's examine the interval x > 1. Here, both factors (x+3) and (x-1) are positive. Their product is also positive. However, the negative sign in front of the factored expression makes the function negative in this region. As an illustration, let's take x = 2. We get f(2) = -(2+3)(2-1) = -(5)(1) = -5, which is negative, thus confirming the statement for x > 1.

Based on this thorough analysis, Statement B is true. The function f(x) = -(x+3)(x-1) is indeed negative for all real values of x where x < -3 and where x > 1. This accurately describes the behavior of the parabola outside the interval between its roots. This conclusion highlights the importance of considering the roots and the leading coefficient when analyzing the sign of a quadratic function.

Conclusion: A Clear Understanding of Quadratic Functions

In conclusion, by meticulously analyzing the function f(x) = -(x+3)(x-1), we have determined that Statement B is the true statement. The function is negative for all real values of x where x < -3 and where x > 1. Statement A, on the other hand, is false, as the function is not positive for all x < -1. This exploration underscores the importance of a comprehensive understanding of quadratic functions, including their roots, vertex, and overall shape, to accurately determine their behavior across different intervals. The ability to analyze functions in this manner is a valuable skill in mathematics and beyond, enabling us to model and understand various real-world phenomena.

This detailed analysis provides a solid foundation for further mathematical explorations. Understanding the behavior of quadratic functions is a crucial step toward tackling more complex mathematical concepts. By mastering these fundamentals, students and enthusiasts alike can confidently approach a wider range of mathematical challenges. Remember, the key to success in mathematics lies in a thorough understanding of the basics and the ability to apply them effectively.