On What Interval Is The Graph Of The Function H(x) = |x-10| + 6 Increasing?

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  • Introduction
  • Understanding Absolute Value Functions
  • Analyzing the Graph of h(x) = |x - 10| + 6
  • Determining the Interval of Increase
  • Step-by-step Solution
  • Graphical Representation
  • Explanation of the Correct Answer
  • Why Other Options are Incorrect
  • Key Takeaways
  • Conclusion

Introduction

In mathematics, understanding the behavior of functions is crucial, and one aspect of this is identifying the intervals where a function is increasing or decreasing. This article delves into the analysis of the absolute value function h(x) = |x - 10| + 6 to determine the interval where its graph is increasing. We will explore the properties of absolute value functions, analyze the given function, and provide a step-by-step explanation to identify the correct interval. This exploration is essential for students and enthusiasts aiming to strengthen their understanding of functions and their graphical representations. The ability to determine intervals of increase and decrease is fundamental in calculus and advanced mathematical studies, making this topic both relevant and important.

Understanding Absolute Value Functions

To effectively analyze the function h(x) = |x - 10| + 6, it's essential to grasp the nature of absolute value functions. The absolute value of a number is its distance from zero on the number line, always resulting in a non-negative value. Mathematically, the absolute value of x, denoted as |x|, is defined as x if x ≥ 0 and -x if x < 0. This definition creates a V-shaped graph for the basic absolute value function, f(x) = |x|, with the vertex at the origin (0, 0). Transformations of this basic function, such as horizontal and vertical shifts, can alter the position and shape of the V, but the fundamental V-shape remains. Understanding how these transformations affect the graph is crucial for analyzing more complex absolute value functions. For example, h(x) = |x - 10| represents a horizontal shift of the basic function by 10 units to the right, and adding 6, as in h(x) = |x - 10| + 6, shifts the entire graph 6 units upward. These transformations directly impact where the function is increasing or decreasing, which we will explore in detail.

Analyzing the Graph of h(x) = |x - 10| + 6

The function h(x) = |x - 10| + 6 is an absolute value function that has been transformed from the basic f(x) = |x|. The term |x - 10| indicates a horizontal shift. Specifically, the graph of |x| is shifted 10 units to the right. This shift places the vertex of the V-shaped graph at x = 10. The addition of +6 outside the absolute value signifies a vertical shift of 6 units upward. Therefore, the vertex of the graph of h(x) is at the point (10, 6). Understanding the vertex is key because it is the point where the function changes direction; to the left of the vertex, the function decreases, and to the right, it increases. This behavior is characteristic of absolute value functions, which exhibit symmetry around their vertex. The graph of h(x) decreases from negative infinity up to x = 10, reaching its minimum value of 6 at the vertex, and then increases from x = 10 towards positive infinity. This distinct change in direction makes the vertex a critical point for analyzing the intervals of increase and decrease.

Determining the Interval of Increase

The question at hand focuses on identifying the interval where the graph of h(x) = |x - 10| + 6 is increasing. As discussed earlier, the graph of an absolute value function in the form |x - a| + b has a V-shape, with the vertex at the point (a, b). For our function, the vertex is at (10, 6). The function decreases to the left of the vertex and increases to the right. Therefore, the interval where h(x) is increasing is to the right of x = 10. In interval notation, this is represented as (10, ∞). This notation signifies all values of x greater than 10. The parentheses indicate that 10 is not included in the interval, as it is the point where the function changes direction (the vertex), and at the vertex, the function is neither increasing nor decreasing. The infinity symbol (∞) denotes that the interval extends indefinitely in the positive direction. Thus, the correct interval of increase is all x values greater than 10, illustrating the function's behavior beyond its minimum point.

Step-by-step Solution

To determine the interval where the graph of h(x) = |x - 10| + 6 is increasing, we follow a step-by-step approach:

  1. Identify the function type: Recognize that h(x) is an absolute value function of the form |x - a| + b.
  2. Find the vertex: The vertex of the graph is at the point (a, b). In this case, a = 10 and b = 6, so the vertex is at (10, 6).
  3. Determine the direction of the V-shape: Since the coefficient of the absolute value term is positive, the V-shape opens upwards. This means the function decreases to the left of the vertex and increases to the right.
  4. Identify the interval of increase: The function increases to the right of the vertex, which corresponds to x values greater than 10.
  5. Express the interval in interval notation: The interval where h(x) is increasing is (10, ∞).

This systematic approach ensures a clear and accurate determination of the interval of increase for the given absolute value function.

Graphical Representation

Visualizing the graph of h(x) = |x - 10| + 6 provides a clear understanding of its behavior. The graph is a V-shaped curve with its vertex at the point (10, 6). To the left of x = 10, the graph slopes downwards, indicating that the function is decreasing in this interval. To the right of x = 10, the graph slopes upwards, demonstrating that the function is increasing. This visual representation directly confirms our analytical determination that the function is increasing for x values greater than 10. By plotting the function, we can observe the symmetry around the vertex and the distinct change in the slope at x = 10. This graphical confirmation is a valuable tool in understanding the behavior of functions, particularly in identifying intervals of increase and decrease. The visual clarity reinforces the mathematical concept and makes it easier to grasp for learners.

Explanation of the Correct Answer

The correct answer to the question,