Graph Of F(x) = |x - H| + K Explained

Understanding the behavior and characteristics of absolute value functions is crucial in mathematics. The function f(x) = |x - h| + k represents a transformed absolute value function, where (h, k) signifies the vertex of the graph. This article delves into the intricacies of this function, specifically focusing on how to determine the values of h and k given certain points and vertex information. We will explore the properties of absolute value functions, the impact of transformations, and a step-by-step approach to solving problems involving this type of function.

Understanding Absolute Value Functions

Before we dive into the specific problem, let's solidify our understanding of absolute value functions. Absolute value functions, at their core, return the magnitude of a number, disregarding its sign. This means that the absolute value of a number is its distance from zero. The most basic absolute value function is f(x) = |x|, which produces a V-shaped graph symmetrical about the y-axis, with its vertex at the origin (0, 0). Understanding this fundamental concept is the key to navigating transformations of absolute value functions.

The graph of f(x) = |x| has two distinct linear sections: one where x ≥ 0, and the function simplifies to f(x) = x, resulting in a line with a slope of 1; and another where x < 0, and the function becomes f(x) = -x, yielding a line with a slope of -1. The point where these two lines meet forms the vertex, which is a crucial point for understanding the function's behavior. This symmetry and the V-shape are the hallmarks of absolute value functions, and any transformations applied to the basic function will maintain these fundamental characteristics.

Now, let's consider the transformed absolute value function f(x) = |x - h| + k. The parameters h and k play pivotal roles in shifting and positioning the graph. The parameter h dictates the horizontal shift. A positive value of h shifts the graph to the right by h units, while a negative value shifts it to the left. The parameter k, on the other hand, controls the vertical shift. A positive k moves the graph upwards by k units, and a negative k shifts it downwards. The combination of these horizontal and vertical shifts determines the location of the vertex, which, in this case, is at the point (h, k). This understanding is crucial for analyzing and interpreting the graph of the function.

Step 1: Finding the Value of h

In the given problem, we have the function f(x) = |x - h| + k, and we are provided with two points on the graph: (-6, -2) and (0, -2). Additionally, we know that the vertex of the graph is at (h, -5). The key to finding h lies in recognizing the symmetry of the absolute value function. The two given points have the same y-coordinate, which means they are equidistant from the vertical line passing through the vertex. This vertical line is the axis of symmetry for the absolute value function, and its x-coordinate is h.

To find the value of h, we can use the midpoint formula. The midpoint of the line segment connecting the two points (-6, -2) and (0, -2) will lie on the axis of symmetry. The midpoint formula is given by [(x1 + x2)/2, (y1 + y2)/2]. Applying this formula to our points, we get:

Midpoint = [(-6 + 0)/2, (-2 + -2)/2] = (-3, -2)

The x-coordinate of the midpoint, -3, represents the x-coordinate of the vertex, which is h. Therefore, we have found that h = -3. This approach leverages the symmetrical nature of absolute value functions to efficiently determine the horizontal shift.

Another way to visualize this is to consider the distance between each point and the line x = h. Since the y-coordinates are the same, the distances along the x-axis must be equal. This means that the value of h is exactly in the middle of -6 and 0 on the number line, which again leads us to h = -3. This understanding of symmetry is a powerful tool in solving problems related to absolute value functions.

By identifying the symmetrical relationship between the given points and the vertex, we were able to determine the value of h without directly using the function's equation. This demonstrates the importance of recognizing geometric properties in mathematical problem-solving. The value of h provides crucial information about the horizontal shift of the graph and is a key component in defining the absolute value function.

Step 2: Determining the Value of k

Now that we have found the value of h to be -3, we can proceed to determine the value of k. We know that the vertex of the graph is at (h, -5), which means the coordinates of the vertex are (-3, -5). From the function f(x) = |x - h| + k, we know that the vertex is also represented by the point (h, k). Therefore, by direct comparison, we can conclude that k = -5. This straightforward approach highlights the significance of the vertex in defining the parameters of the absolute value function.

Alternatively, we can use one of the given points, say (0, -2), and substitute the known values of x, f(x), and h into the function's equation to solve for k. The equation becomes:

-2 = |0 - (-3)| + k

Simplifying this equation, we get:

-2 = |3| + k

-2 = 3 + k

Subtracting 3 from both sides, we find:

k = -5

This confirms our previous finding that k = -5. Using this method, we are essentially applying the definition of the function to a specific point on the graph to extract the value of k. This approach demonstrates the connection between the function's equation and its graphical representation.

Similarly, we could have used the point (-6, -2) to solve for k. Substituting these values into the function, we get:

-2 = |-6 - (-3)| + k

Simplifying, we have:

-2 = |-3| + k

-2 = 3 + k

And again, we find that k = -5. The consistency of the result, regardless of which point is used, reinforces the accuracy of our solution and the fundamental properties of the function.

The value of k represents the vertical shift of the graph and determines the minimum value of the function, which occurs at the vertex. In this case, the graph is shifted downwards by 5 units. Understanding the role of k is essential for sketching the graph and analyzing the function's behavior.

Step 3: Complete Function and Graph Interpretation

With the values of h and k determined, we can now write the complete equation of the function: f(x) = |x - (-3)| + (-5), which simplifies to f(x) = |x + 3| - 5. This equation fully describes the absolute value function, and we can use it to analyze the graph's characteristics.

The vertex of the graph is located at (-3, -5). This is the lowest point on the graph, and it serves as the turning point for the V-shape. The graph opens upwards because the coefficient of the absolute value term is positive (1 in this case). This means that the function has a minimum value of -5, which occurs at x = -3. Identifying the vertex is a critical first step in understanding the graph's behavior.

The axis of symmetry for the graph is the vertical line x = -3. This line divides the graph into two symmetrical halves. Understanding the axis of symmetry helps in visualizing the graph's shape and predicting its behavior.

To further analyze the graph, we can consider the slopes of the two linear sections. For x > -3, the function behaves like f(x) = (x + 3) - 5 = x - 2, which has a slope of 1. For x < -3, the function behaves like f(x) = -(x + 3) - 5 = -x - 8, which has a slope of -1. The slopes of these linear sections define the steepness of the V-shape.

The y-intercept of the graph can be found by setting x = 0 in the equation: f(0) = |0 + 3| - 5 = 3 - 5 = -2. This confirms that the point (0, -2) lies on the graph, as given in the problem. Finding intercepts provides additional key points for sketching the graph accurately.

The x-intercepts can be found by setting f(x) = 0 and solving for x: 0 = |x + 3| - 5. This leads to two equations: x + 3 = 5 and x + 3 = -5. Solving these equations, we get x = 2 and x = -8. Thus, the x-intercepts are at (2, 0) and (-8, 0). Knowing the x-intercepts provides a complete picture of where the graph crosses the x-axis.

By combining all this information, we can accurately sketch the graph of f(x) = |x + 3| - 5. The graph is a V-shaped curve with a vertex at (-3, -5), symmetrical about the line x = -3, and passing through the points (2, 0), (-8, 0), and (0, -2). This comprehensive analysis demonstrates how to fully understand and interpret the graph of a transformed absolute value function.

Conclusion

In this article, we have thoroughly explored how to find the values of h and k in the absolute value function f(x) = |x - h| + k, given certain points on the graph and the vertex. We emphasized the importance of understanding the symmetry of absolute value functions and how transformations affect the graph. By using the midpoint formula, substituting values into the function's equation, and analyzing the graph's characteristics, we were able to solve the problem effectively.

Mastering the concepts presented in this article is crucial for success in mathematics, particularly in algebra and calculus. The ability to analyze and interpret absolute value functions is a fundamental skill that opens the door to more advanced topics. By applying these principles, you can confidently tackle similar problems and gain a deeper understanding of mathematical functions and their graphical representations.