Analyzing The Graph And Properties Of F(x) = (x-4)(x+1)
This article delves into the analysis of the quadratic function f(x) = (x-4)(x+1), exploring its graphical representation and key characteristics. Understanding the behavior of functions is crucial in mathematics, especially when dealing with quadratic equations and their applications. We will examine the graph of this specific function to determine intervals of increase and decrease, identify critical points, and discuss its overall behavior. Our primary focus will be on dissecting the function's properties directly from its graph, allowing us to make accurate statements about its nature. Specifically, we aim to address the question of whether the function is increasing for all real values of x < 0, a common point of inquiry when analyzing functions.
Understanding Quadratic Functions
To fully grasp the properties of f(x) = (x-4)(x+1), it's important to first understand the general form and characteristics of quadratic functions. A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards depending on the sign of the leading coefficient a. If a > 0, the parabola opens upwards, indicating a minimum value, and if a < 0, it opens downwards, indicating a maximum value. The vertex of the parabola is the point where the function reaches its minimum or maximum value, and it plays a crucial role in determining the intervals of increase and decrease. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
The roots, or x-intercepts, of a quadratic function are the values of x for which f(x) = 0. These roots can be found by solving the quadratic equation ax^2 + bx + c = 0, either by factoring, completing the square, or using the quadratic formula. The roots represent the points where the parabola intersects the x-axis. The y-intercept of the function is the value of f(0), which is the point where the parabola intersects the y-axis. Understanding these basic concepts is essential for analyzing the behavior of any quadratic function, including our specific function f(x) = (x-4)(x+1). By examining the coefficients, roots, and vertex, we can gain valuable insights into the function's graph and properties, allowing us to make informed statements about its increasing and decreasing intervals.
Analyzing the Function f(x) = (x-4)(x+1)
Let's delve into the specifics of the function f(x) = (x-4)(x+1). This function is given in factored form, which provides immediate insights into its roots. The roots are the values of x that make the function equal to zero. Setting f(x) = 0, we have (x-4)(x+1) = 0, which yields the roots x = 4 and x = -1. These are the points where the graph of the function intersects the x-axis. To further analyze the function, we can expand it into the standard quadratic form: f(x) = x^2 - 3x - 4. This form allows us to identify the coefficients: a = 1, b = -3, and c = -4. Since a = 1, which is positive, the parabola opens upwards, indicating that the function has a minimum value.
The vertex of the parabola is a crucial point for determining the intervals of increase and decrease. The x-coordinate of the vertex can be found using the formula x = -b / 2a. In this case, x = -(-3) / (2 * 1) = 3/2 = 1.5. To find the y-coordinate of the vertex, we substitute this value back into the function: f(1.5) = (1.5 - 4)(1.5 + 1) = (-2.5)(2.5) = -6.25. Therefore, the vertex of the parabola is at the point (1.5, -6.25). This is the minimum point of the function. The axis of symmetry is the vertical line passing through the vertex, which is x = 1.5. Now, we can analyze the increasing and decreasing intervals. Since the parabola opens upwards, the function is decreasing to the left of the vertex and increasing to the right of the vertex. This means that the function is decreasing for x < 1.5 and increasing for x > 1.5. This understanding of the function's behavior is essential for evaluating the given statement about the function's increasing nature for x < 0.
Determining Intervals of Increase and Decrease
The concept of increasing and decreasing intervals is fundamental in understanding the behavior of a function. A function is said to be increasing on an interval if its values increase as the input variable x increases within that interval. Conversely, a function is decreasing on an interval if its values decrease as x increases. For quadratic functions, these intervals are directly related to the vertex of the parabola. As we established earlier, the function f(x) = (x-4)(x+1) has a vertex at (1.5, -6.25) and opens upwards. This means that the function decreases as we move from left to right until we reach the vertex, and then it starts increasing as we continue moving right.
Specifically, the function is decreasing for all x values less than the x-coordinate of the vertex, which is 1.5. In mathematical notation, this interval is (-∞, 1.5). On the other hand, the function is increasing for all x values greater than the x-coordinate of the vertex, which is 1.5. This interval is represented as (1.5, ∞). These intervals are crucial for analyzing the statement about the function's behavior for x < 0. To determine whether the function is increasing for all real values of x < 0, we need to consider the interval (-∞, 0). Since this interval is a subset of the decreasing interval (-∞, 1.5), we can conclude that the function is not increasing for all x < 0. In fact, the function is decreasing in this interval. This detailed analysis of increasing and decreasing intervals, derived from the graph and the vertex, allows us to accurately evaluate the given statement about the function's properties.
Evaluating the Statement: Increasing for x < 0
Now, let's directly address the statement: "The function is increasing for all real values of x where x < 0." As we established in the previous section, the function f(x) = (x-4)(x+1) is decreasing for x < 1.5. The interval x < 0 is a subset of the interval x < 1.5. Therefore, if the function is decreasing for all x values less than 1.5, it must also be decreasing for all x values less than 0. This directly contradicts the statement that the function is increasing for x < 0.
To further illustrate this, consider a few specific values of x less than 0. For example, let's take x = -2 and x = -3. Evaluating the function at these points, we have: f(-2) = (-2 - 4)(-2 + 1) = (-6)(-1) = 6 and f(-3) = (-3 - 4)(-3 + 1) = (-7)(-2) = 14. As x decreases from -2 to -3, the function value increases from 6 to 14. This might initially suggest that the function is increasing. However, if we consider values closer to the root at x = -1, such as x = -0.5 and x = -0.75, we find: f(-0.5) = (-0.5 - 4)(-0.5 + 1) = (-4.5)(0.5) = -2.25 and f(-0.75) = (-0.75 - 4)(-0.75 + 1) = (-4.75)(0.25) = -1.1875. Here, as x decreases from -0.5 to -0.75, the function value decreases from -2.25 to -1.1875, indicating that the function is decreasing. This inconsistency reinforces the fact that the function is not increasing for all x < 0.
Conclusion: The Function's Behavior
In conclusion, after a thorough analysis of the quadratic function f(x) = (x-4)(x+1), we can confidently state that the function is not increasing for all real values of x where x < 0. Our examination of the graph, the vertex, and the intervals of increase and decrease clearly demonstrates that the function is decreasing in the interval (-∞, 1.5), which includes all x values less than 0. This comprehensive analysis highlights the importance of understanding the fundamental properties of quadratic functions, including their roots, vertex, and intervals of monotonicity, to accurately describe their behavior. By combining graphical analysis with algebraic techniques, we can gain valuable insights into the characteristics of these functions and make informed statements about their nature.
This exploration of f(x) = (x-4)(x+1) serves as a prime example of how mathematical analysis can provide a clear and precise understanding of function behavior. The ability to identify increasing and decreasing intervals is a crucial skill in calculus and other advanced mathematical fields, making this type of analysis essential for students and professionals alike. Therefore, mastering the techniques discussed in this article is vital for anyone seeking a deeper understanding of functions and their properties.
