Analyzing The Graph Of F(x) = (x+6)(x+2) Intervals, Symmetry, And Behavior

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In the realm of mathematics, quadratic functions hold a significant position, exhibiting a unique parabolic shape that intrigues mathematicians and students alike. Understanding the properties and behavior of these functions is crucial for various applications, from physics to engineering to economics. In this article, we will delve into the intricacies of a specific quadratic function, f(x) = (x+6)(x+2), analyzing its graph and uncovering its key characteristics. We will explore its roots, axis of symmetry, intervals of positivity and negativity, and intervals of increasing and decreasing behavior. By carefully examining these aspects, we will gain a comprehensive understanding of the function's graphical representation and its mathematical essence.

1. The Negative Interval: Identifying the Region Below the x-axis

One of the fundamental aspects of understanding a function's behavior is determining the intervals over which it assumes negative values. In other words, we seek to identify the regions on the x-axis where the graph of the function lies below the x-axis. For the function f(x) = (x+6)(x+2), the function is negative when the product of the two factors, (x+6) and (x+2), is negative. This occurs when one factor is positive and the other is negative. Let's analyze the intervals where this condition holds:

  • When x < -6: Both (x+6) and (x+2) are negative, resulting in a positive product. Therefore, the function is not negative in this interval.
  • When -6 < x < -2: (x+6) is positive, and (x+2) is negative, resulting in a negative product. Thus, the function is negative in this interval.
  • When x > -2: Both (x+6) and (x+2) are positive, resulting in a positive product. Hence, the function is not negative in this interval.

Based on this analysis, we can confidently conclude that the function f(x) = (x+6)(x+2) is negative over the interval (-6, -2). This means that the portion of the graph between x = -6 and x = -2 lies below the x-axis. This understanding is crucial for solving inequalities and analyzing real-world scenarios modeled by quadratic functions.

2. The Axis of Symmetry: Pinpointing the Parabola's Center

The axis of symmetry is a crucial characteristic of a parabola, representing the vertical line that divides the parabola into two symmetrical halves. It essentially acts as a mirror, reflecting one half of the parabola onto the other. For a quadratic function in the standard form f(x) = ax^2 + bx + c, the axis of symmetry is given by the equation x = -b / 2a. However, for the function f(x) = (x+6)(x+2), we can determine the axis of symmetry more directly by recognizing that it lies exactly midway between the roots of the quadratic equation.

The roots of the equation f(x) = (x+6)(x+2) = 0 are the values of x that make the function equal to zero. These roots are x = -6 and x = -2. The axis of symmetry lies exactly in the middle of these two roots. To find the midpoint, we simply calculate the average of the roots:

  • Midpoint = (-6 + (-2)) / 2 = -8 / 2 = -4

Therefore, the axis of symmetry for the function f(x) = (x+6)(x+2) is the vertical line x = -4. This line passes through the vertex of the parabola, which is the point where the parabola changes direction. The axis of symmetry is a fundamental element in understanding the symmetry and behavior of the parabolic graph.

3. Intervals of Increasing Behavior: Tracing the Ascent of the Parabola

Another essential aspect of analyzing a function's graph is identifying the intervals over which the function is increasing or decreasing. A function is said to be increasing over an interval if its values increase as the input values (x) increase within that interval. For a parabola, the function is increasing on one side of the vertex and decreasing on the other side. The vertex is the turning point of the parabola, where it changes direction.

For the function f(x) = (x+6)(x+2), we know that the axis of symmetry is x = -4. The vertex of the parabola lies on this line. Since the coefficient of the x^2 term in the expanded form of the function is positive (1), the parabola opens upwards. This means that the function decreases to the left of the vertex and increases to the right of the vertex.

To find the exact interval of increasing behavior, we need to determine the x-coordinate of the vertex. We already know that the x-coordinate of the vertex is -4 (the axis of symmetry). Therefore, the function f(x) = (x+6)(x+2) is increasing over the interval (-4, ∞). This means that as x increases from -4 towards positive infinity, the function values also increase. Understanding the intervals of increasing and decreasing behavior is vital for optimizing functions and solving related problems.

In this comprehensive analysis, we have successfully explored the graph of the function f(x) = (x+6)(x+2), uncovering its essential characteristics. We have determined that the function is negative over the interval (-6, -2), indicating the region where the graph lies below the x-axis. We have also pinpointed the axis of symmetry as the vertical line x = -4, representing the parabola's central line of reflection. Furthermore, we have identified the interval of increasing behavior as (-4, ∞), signifying the region where the function values increase as x increases.

By understanding these key descriptors, we gain a profound appreciation for the behavior and graphical representation of the quadratic function f(x) = (x+6)(x+2). This knowledge not only enhances our mathematical comprehension but also equips us with the tools to analyze and solve real-world problems that can be modeled using quadratic functions. The ability to interpret and analyze graphs is a fundamental skill in mathematics and its applications. This exploration serves as a stepping stone for further investigations into the fascinating world of functions and their graphical representations.

The graph of the function f(x) = (x+6)(x+2) is shown. Which three of the following statements accurately describe the graph's characteristics?

  • The function yields negative values within the interval (-6, -2).
  • The axis of symmetry is the vertical line x = -4.
  • The function's values increase as x increases within the interval (-4, ∞).

Graph Analysis of f(x) = (x+6)(x+2) | Negative Intervals, Symmetry, and Increasing Behavior