Which Equation Has Vertex (-6, 7) An Absolute Value Exploration

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In mathematics, absolute value functions play a crucial role in various applications, from simple algebraic manipulations to complex mathematical modeling. At the heart of understanding these functions lies the concept of the vertex, which is the point where the graph changes direction. This article delves into the world of absolute value equations, focusing on how to identify the vertex and construct equations that match a specific vertex. We will explore the general form of absolute value equations, the significance of the vertex, and how to determine the equation given a vertex. By the end of this discussion, you'll be equipped with the knowledge to confidently tackle problems involving absolute value functions and their vertices.

Decoding Absolute Value Equations

To identify the equation with a given vertex, we must first understand the general form of an absolute value equation. An absolute value equation typically takes the form:

y=a∣x−h∣+ky = a|x - h| + k

Where:

  • y represents the dependent variable.
  • x represents the independent variable.
  • a determines the steepness and direction of the graph. If a is positive, the graph opens upwards, and if a is negative, it opens downwards. The larger the absolute value of a, the steeper the graph.
  • (h, k) represents the vertex of the absolute value function. This is the point where the graph changes direction, forming a “V” shape. The vertex is a critical point for understanding the behavior and characteristics of the absolute value function.

Understanding the roles of a, h, and k is paramount to manipulating and interpreting absolute value equations. The a value dictates the vertical stretch or compression and the reflection across the x-axis. The h value represents the horizontal shift of the graph, and the k value represents the vertical shift. By recognizing these transformations, we can easily determine how the graph of the absolute value function is altered from the basic form y = |x|.

The Significance of the Vertex

The vertex of an absolute value function is arguably its most crucial feature. It represents the point at which the function transitions from decreasing to increasing (if a is positive) or from increasing to decreasing (if a is negative). In the context of the equation $y = a|x - h| + k$, the vertex is located at the point (h, k). The x-coordinate, h, indicates the horizontal shift from the origin, and the y-coordinate, k, indicates the vertical shift.

The vertex serves as the point of symmetry for the graph of the absolute value function. The two sides of the “V” shape are mirror images of each other across a vertical line passing through the vertex. This symmetry is a fundamental property of absolute value functions and can be used to simplify graphing and problem-solving. Moreover, the vertex is essential for determining the range of the function. If a is positive, the range is all y values greater than or equal to k, and if a is negative, the range is all y values less than or equal to k.

Identifying the Correct Equation

Now, let's apply this knowledge to the question at hand: Which of the following equations has the vertex $(-6, 7)$?

We are looking for an equation in the form $y = a|x - h| + k$ where h = -6 and k = 7. Substituting these values into the general form, we get:

y=a∣x−(−6)∣+7y = a|x - (-6)| + 7

Simplifying, we have:

y=a∣x+6∣+7y = a|x + 6| + 7

Comparing this with the given options, we can identify the correct equation by matching the form.

Analyzing the Options

Let's examine each option provided in the question to determine which one has the vertex $(-6, 7)$. The options are:

A. y=6∣x+5∣−7y = 6|x + 5| - 7
B. y=−4∣x−6∣−7y = -4|x - 6| - 7
C. y=−5∣x+6∣+7y = -5|x + 6| + 7
D. y=6∣x+4∣−7y = 6|x + 4| - 7

We need to rewrite each equation in the form $y = a|x - h| + k$ and identify the vertex (h, k).

Option A: y=6∣x+5∣−7y = 6|x + 5| - 7

For option A, the equation is y=6∣x+5∣−7y = 6|x + 5| - 7. Comparing this to the general form, we can identify:

  • a = 6
  • h = -5 (since x+5x + 5 can be written as x−(−5)x - (-5))
  • k = -7

Thus, the vertex for option A is $(-5, -7)$, which does not match the desired vertex of $(-6, 7)$.

Option B: y=−4∣x−6∣−7y = -4|x - 6| - 7

For option B, the equation is y=−4∣x−6∣−7y = -4|x - 6| - 7. Here, we have:

  • a = -4
  • h = 6
  • k = -7

The vertex for option B is $(6, -7)$, which also does not match the target vertex of $(-6, 7)$.

Option C: y=−5∣x+6∣+7y = -5|x + 6| + 7

Option C presents the equation y=−5∣x+6∣+7y = -5|x + 6| + 7. Identifying the parameters:

  • a = -5
  • h = -6 (since x+6x + 6 is x−(−6)x - (-6))
  • k = 7

The vertex for option C is $(-6, 7)$, which exactly matches the desired vertex. Therefore, option C is the correct answer.

Option D: y=6∣x+4∣−7y = 6|x + 4| - 7

Finally, let's consider option D, y=6∣x+4∣−7y = 6|x + 4| - 7. The parameters are:

  • a = 6
  • h = -4 (since x+4x + 4 is x−(−4)x - (-4))
  • k = -7

The vertex for option D is $(-4, -7)$, which does not align with the desired vertex of $(-6, 7)$.

Conclusion: Identifying the Correct Equation

After analyzing all the options, it is evident that option C, y=−5∣x+6∣+7y = -5|x + 6| + 7, is the only equation with the vertex $(-6, 7)$. This determination was made by understanding the general form of absolute value equations and correctly identifying the values of h and k that define the vertex.

In summary, mastering the concept of the vertex and its role in absolute value equations is essential for solving mathematical problems efficiently. By understanding how the parameters a, h, and k influence the graph of an absolute value function, you can confidently identify equations that meet specific criteria, such as having a particular vertex. The process involves rewriting equations in the standard form, identifying the h and k values, and comparing them with the desired vertex coordinates. With practice, this skill becomes invaluable for tackling more complex mathematical challenges.

Further Exploration of Absolute Value Functions

To deepen your understanding of absolute value functions, consider exploring the following topics:

  1. Graphing Absolute Value Functions: Learn how to accurately graph absolute value functions by plotting points, using transformations, and identifying key features like the vertex and intercepts.
  2. Solving Absolute Value Equations: Practice solving equations involving absolute values by isolating the absolute value expression and considering both positive and negative cases.
  3. Absolute Value Inequalities: Extend your knowledge to solving inequalities containing absolute values, understanding how to interpret the solutions in terms of intervals on the number line.
  4. Applications of Absolute Value: Explore real-world applications of absolute value functions, such as in distance calculations, error analysis, and modeling symmetrical phenomena.

By delving into these areas, you'll gain a comprehensive understanding of absolute value functions and their versatility in mathematical problem-solving. Remember, the key to mastering mathematics lies in consistent practice and a willingness to explore new concepts and applications. Happy learning!