Decreasing Interval Of F(x) = 2(x+3)^2 + 2 Explained

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#Consider the graph of the function f(x) = 2(x+3)^2 + 2. Over which interval is the graph decreasing?

Determining the intervals where a function is increasing or decreasing is a fundamental concept in calculus and precalculus mathematics. This analysis helps us understand the behavior and shape of a function's graph. In this article, we will delve into how to identify the decreasing interval of the given quadratic function, f(x) = 2(x+3)^2 + 2. We will break down the characteristics of quadratic functions, focusing on how their vertex form provides critical information about their increasing and decreasing behavior.

Analyzing Quadratic Functions

Quadratic functions, which are polynomials of degree two, have a distinctive U-shaped graph known as a parabola. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. The sign of the leading coefficient, a, determines the parabola's orientation: if a > 0, the parabola opens upwards, and if a < 0, it opens downwards. This concavity significantly impacts the function's increasing and decreasing intervals.

The vertex form of a quadratic function, expressed as f(x) = a(x - h)^2 + k, provides a more direct way to identify key features of the parabola. In this form, (h, k) represents the vertex of the parabola, which is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value. The axis of symmetry is the vertical line x = h, which divides the parabola into two symmetrical halves. Understanding these elements is crucial for determining where the function is increasing or decreasing.

In our specific case, the function f(x) = 2(x+3)^2 + 2 is already given in vertex form. By comparing it to the general vertex form f(x) = a(x - h)^2 + k, we can easily identify the parameters: a = 2, h = -3, and k = 2. Since a = 2 is positive, the parabola opens upwards, indicating that the function has a minimum value at the vertex. The vertex of this parabola is therefore at the point (-3, 2). This knowledge is the cornerstone for determining the intervals of increasing and decreasing behavior.

Identifying the Decreasing Interval

To determine the decreasing interval of the function f(x) = 2(x+3)^2 + 2, we need to consider the shape of the parabola and the location of its vertex. Since the parabola opens upwards (because a = 2 is positive), the function decreases as we move from left to right along the x-axis until we reach the vertex. After the vertex, the function starts to increase. The vertex, therefore, marks the turning point where the function transitions from decreasing to increasing.

The x-coordinate of the vertex, h, is the critical value that defines the interval where the function is decreasing. In this case, the vertex is at (-3, 2), so h = -3. The function decreases for all x-values less than -3. This means the decreasing interval is from negative infinity up to -3. In interval notation, this is represented as (-∞, -3).

It's important to include the endpoint -3 in our consideration. At x = -3, which is the x-coordinate of the vertex, the function momentarily stops decreasing and begins to increase. Therefore, the interval where the function is decreasing includes all values strictly less than -3, but not -3 itself. However, in some contexts, it's conventional to include the endpoint in the interval if the function is considered decreasing up to that point. In this context, we include -3, denoting the decreasing interval as (-∞, -3]. This notation indicates that the function is decreasing up to and including the point x = -3.

Visualizing the graph of the function can further clarify this concept. Imagine a U-shaped parabola with its vertex at (-3, 2). As you trace the graph from left to right, you’ll notice that the y-values decrease until you reach the vertex. After the vertex, the y-values increase. This visual representation vividly illustrates the decreasing behavior of the function on the interval (-∞, -3].

Step-by-Step Solution

To solidify our understanding, let’s walk through a step-by-step solution to identify the decreasing interval of f(x) = 2(x+3)^2 + 2:

  1. Identify the vertex form: The function is already in vertex form, f(x) = a(x - h)^2 + k, where a = 2, h = -3, and k = 2.
  2. Determine the vertex: The vertex of the parabola is (h, k) = (-3, 2).
  3. Determine the direction of the parabola: Since a = 2 is positive, the parabola opens upwards.
  4. Identify the decreasing interval: For a parabola that opens upwards, the function decreases to the left of the vertex. Thus, the decreasing interval is (-∞, -3].

This step-by-step process provides a clear and logical approach to solving similar problems. By recognizing the significance of the vertex form and understanding how the leading coefficient affects the shape of the parabola, you can easily determine the decreasing intervals of quadratic functions.

Graphical Representation and Interpretation

The graphical representation of f(x) = 2(x+3)^2 + 2 offers an intuitive way to understand its decreasing interval. When we plot this function, we see a parabola that opens upwards with its vertex at (-3, 2). The left side of the parabola (to the left of x = -3) slopes downwards, indicating that the function values decrease as x increases. The right side of the parabola (to the right of x = -3) slopes upwards, showing that the function values increase as x increases.

This graphical behavior directly corresponds to the decreasing and increasing intervals we identified analytically. The decreasing interval (-∞, -3] is visually represented by the downward-sloping portion of the parabola to the left of its vertex. The increasing interval, which we haven't explicitly discussed yet but is equally important, is represented by the upward-sloping portion of the parabola to the right of its vertex, corresponding to the interval [-3, ∞). The vertex at (-3, 2) serves as the pivotal point where the function transitions from decreasing to increasing.

Using graphing tools or software can further enhance our understanding. By plotting the function, we can dynamically observe how changes in the coefficients a, h, and k affect the parabola's shape and position, and consequently, its decreasing and increasing intervals. For instance, changing the value of a will either stretch or compress the parabola vertically, but the vertex's x-coordinate will remain the same, thereby not affecting the decreasing interval. However, if we change h, the vertex shifts horizontally, which directly alters the decreasing and increasing intervals. Understanding these transformations is crucial for a comprehensive grasp of quadratic functions.

The Role of the Derivative

In calculus, the derivative of a function provides a powerful tool for determining intervals of increase and decrease. The derivative, denoted as f'(x), gives the instantaneous rate of change of the function f(x) with respect to x. If f'(x) > 0 over an interval, the function is increasing on that interval. Conversely, if f'(x) < 0 over an interval, the function is decreasing on that interval. If f'(x) = 0, the function has a critical point, which could be a local minimum, local maximum, or a point of inflection.

For the quadratic function f(x) = 2(x+3)^2 + 2, we can find the derivative using the power rule and chain rule. First, expand the function: f(x) = 2(x^2 + 6x + 9) + 2 = 2x^2 + 12x + 18 + 2 = 2x^2 + 12x + 20. Now, differentiate with respect to x: f'(x) = 4x + 12.

To find the critical points, we set f'(x) = 0: 4x + 12 = 0. Solving for x, we get x = -3. This critical point corresponds to the x-coordinate of the vertex we found earlier. Now, we analyze the sign of f'(x) in the intervals determined by this critical point:

  • For x < -3, let’s test x = -4: f'(-4) = 4(-4) + 12 = -16 + 12 = -4, which is negative. Thus, the function is decreasing on the interval (-∞, -3).
  • For x > -3, let’s test x = -2: f'(-2) = 4(-2) + 12 = -8 + 12 = 4, which is positive. Thus, the function is increasing on the interval (-3, ∞).

This calculus-based approach confirms our earlier findings using the properties of the vertex form. The derivative provides a rigorous method for determining intervals of increase and decrease, and it’s particularly useful for functions that are not in a standard form like the vertex form of a quadratic.

Common Mistakes and How to Avoid Them

When determining the decreasing intervals of quadratic functions, several common mistakes can lead to incorrect answers. Recognizing these pitfalls and understanding how to avoid them is crucial for mastering this concept. One frequent mistake is confusing the decreasing interval with the increasing interval. This often happens when students don’t fully grasp the relationship between the parabola’s orientation (whether it opens upwards or downwards) and its increasing/decreasing behavior.

Another common error is misinterpreting the vertex form of the quadratic function. Students might incorrectly identify the h and k values, or they might not realize that the vertex is the turning point where the function changes from decreasing to increasing (or vice versa). A clear understanding of the vertex form f(x) = a(x - h)^2 + k and how the parameters a, h, and k affect the parabola's shape and position is essential.

A third mistake involves incorrectly using interval notation. For instance, students might use parentheses instead of brackets, or vice versa, leading to an incorrect representation of the interval. Remember that parentheses indicate that the endpoint is not included in the interval, while brackets indicate that the endpoint is included. In the context of decreasing intervals, it’s generally accepted to include the endpoint if the function is considered decreasing up to that point.

To avoid these mistakes, it’s helpful to follow a systematic approach:

  1. Identify the form of the function: Is it in general form or vertex form? If it's in general form, consider converting it to vertex form to easily identify the vertex.
  2. Determine the vertex: Find the coordinates of the vertex (h, k). This is the turning point of the parabola.
  3. Determine the direction of the parabola: Is the leading coefficient a positive or negative? This determines whether the parabola opens upwards or downwards.
  4. Identify the decreasing and increasing intervals: For a parabola that opens upwards, the function decreases to the left of the vertex and increases to the right. For a parabola that opens downwards, the function increases to the left of the vertex and decreases to the right.
  5. Use correct interval notation: Use parentheses and brackets appropriately to represent the intervals.

Regular practice and careful attention to detail are key to avoiding these common mistakes. By thoroughly understanding the properties of quadratic functions and following a structured approach, you can confidently determine their decreasing intervals.

Conclusion

In conclusion, determining the decreasing interval of the quadratic function f(x) = 2(x+3)^2 + 2 involves understanding the properties of parabolas and their vertex form. By identifying the vertex as (-3, 2) and recognizing that the parabola opens upwards, we can conclude that the function is decreasing on the interval (-∞, -3]. This analysis is crucial for understanding the behavior of quadratic functions and their graphical representation.

We explored the concept from multiple angles, including the properties of quadratic functions, the significance of the vertex form, graphical interpretation, the role of the derivative, and common mistakes to avoid. Each of these perspectives reinforces the importance of a systematic approach and a solid understanding of the underlying principles.

Mastering the skill of identifying increasing and decreasing intervals is not only fundamental in mathematics but also has applications in various fields such as physics, engineering, and economics. The ability to analyze functions and understand their behavior is a valuable asset in problem-solving and decision-making. Therefore, a thorough understanding of these concepts is essential for academic and professional success.