Which Function Has A Vertex At { (2,6)$}$?A. { F(x)=2|x+2|-6$}$B. { F(x)=2|x-2|-6$}$C. { F(x)=2|x+2|+6$}$D. { F(x)=2|x-2|+6$}$
Identifying the Function with a Vertex at (2,6)
In mathematics, particularly in algebra and geometry, a vertex is a point where a curve changes direction. When dealing with functions, identifying the vertex of a graph is crucial for understanding its behavior and properties. In this article, we will explore the concept of a vertex and determine which function has a vertex at (2,6).
Understanding the Concept of a Vertex
A vertex is a point on a curve where the curve changes direction. It is a critical point that can be used to describe the shape and behavior of a function. In the context of a graph, the vertex is the lowest or highest point on the curve, depending on the direction of the curve.
Identifying the Vertex of a Function
To identify the vertex of a function, we need to analyze its equation. The general form of a function with a vertex is:
f(x) = a|x - h| + k
where (h, k) is the vertex of the function.
Analyzing the Options
Let's analyze the options given:
A. f(x) = 2|x + 2| - 6 B. f(x) = 2|x - 2| - 6 C. f(x) = 2|x + 2| + 6 D. f(x) = 2|x - 2| + 6
We need to determine which of these functions has a vertex at (2,6).
Option A: f(x) = 2|x + 2| - 6
The equation of this function is f(x) = 2|x + 2| - 6. To find the vertex, we need to identify the values of h and k. In this case, h = -2 and k = -6. However, the vertex is given as (2,6), which does not match the vertex of this function.
Option B: f(x) = 2|x - 2| - 6
The equation of this function is f(x) = 2|x - 2| - 6. To find the vertex, we need to identify the values of h and k. In this case, h = 2 and k = -6. However, the vertex is given as (2,6), which does not match the vertex of this function.
Option C: f(x) = 2|x + 2| + 6
The equation of this function is f(x) = 2|x + 2| + 6. To find the vertex, we need to identify the values of h and k. In this case, h = -2 and k = 6. However, the vertex is given as (2,6), which does not match the vertex of this function.
Option D: f(x) = 2|x - 2| + 6
The equation of this function is f(x) = 2|x - 2| + 6. To find the vertex, we need to identify the values of h and k. In this case, h = 2 and k = 6. This matches the given vertex (2,6).
Conclusion
Based on the analysis, the function with a vertex at (2,6) is:
D. f(x) = 2|x - 2| + 6
This function has a vertex at (2,6), which matches the given vertex.
Understanding the Properties of the Function
The function f(x) = 2|x - 2| + 6 is a type of absolute value function. It has a vertex at (2,6) and is symmetric about the line x = 2. The function has a minimum value of 6 at x = 2 and increases as we move away from x = 2.
Graphing the Function
To graph the function, we can use the following steps:
- Plot the point (2,6) on the graph.
- Draw a line through the point (2,6) that is symmetric about the line x = 2.
- The graph of the function will be a V-shaped graph with the vertex at (2,6).
Real-World Applications
The concept of a vertex is used in various real-world applications, such as:
- Optimization: In optimization problems, the vertex of a function represents the maximum or minimum value of the function.
- Graphing: In graphing, the vertex of a function represents the point where the function changes direction.
- Statistics: In statistics, the vertex of a function represents the mean or median of a dataset.
Conclusion
In conclusion, the function with a vertex at (2,6) is f(x) = 2|x - 2| + 6. This function has a vertex at (2,6) and is symmetric about the line x = 2. The concept of a vertex is used in various real-world applications, such as optimization, graphing, and statistics.
Q&A: Understanding the Function with a Vertex at (2,6)
In our previous article, we explored the concept of a vertex and identified the function with a vertex at (2,6). In this article, we will answer some frequently asked questions about the function and its properties.
Q: What is the vertex of the function f(x) = 2|x - 2| + 6?
A: The vertex of the function f(x) = 2|x - 2| + 6 is (2,6).
Q: What is the equation of the function with a vertex at (2,6)?
A: The equation of the function with a vertex at (2,6) is f(x) = 2|x - 2| + 6.
Q: What is the x-coordinate of the vertex of the function f(x) = 2|x - 2| + 6?
A: The x-coordinate of the vertex of the function f(x) = 2|x - 2| + 6 is 2.
Q: What is the y-coordinate of the vertex of the function f(x) = 2|x - 2| + 6?
A: The y-coordinate of the vertex of the function f(x) = 2|x - 2| + 6 is 6.
Q: Is the function f(x) = 2|x - 2| + 6 symmetric about the line x = 2?
A: Yes, the function f(x) = 2|x - 2| + 6 is symmetric about the line x = 2.
Q: What is the minimum value of the function f(x) = 2|x - 2| + 6?
A: The minimum value of the function f(x) = 2|x - 2| + 6 is 6, which occurs at x = 2.
Q: How do I graph the function f(x) = 2|x - 2| + 6?
A: To graph the function f(x) = 2|x - 2| + 6, plot the point (2,6) on the graph and draw a line through the point that is symmetric about the line x = 2.
Q: What are some real-world applications of the concept of a vertex?
A: The concept of a vertex is used in various real-world applications, such as optimization, graphing, and statistics.
Q: Can I use the concept of a vertex to solve optimization problems?
A: Yes, the concept of a vertex can be used to solve optimization problems. The vertex of a function represents the maximum or minimum value of the function.
Q: How do I find the vertex of a function?
A: To find the vertex of a function, identify the values of h and k in the equation f(x) = a|x - h| + k. The vertex is then given by the point (h, k).
Q: What is the significance of the vertex in graphing?
A: The vertex of a function represents the point where the function changes direction. It is a critical point that can be used to describe the shape and behavior of a function.
Q: Can I use the concept of a vertex to analyze the behavior of a function?
A: Yes, the concept of a vertex can be used to analyze the behavior of a function. The vertex represents the point where the function changes direction, and it can be used to describe the shape and behavior of the function.
Conclusion
In conclusion, the function with a vertex at (2,6) is f(x) = 2|x - 2| + 6. This function has a vertex at (2,6) and is symmetric about the line x = 2. The concept of a vertex is used in various real-world applications, such as optimization, graphing, and statistics.