Which Functions Have A Vertex With An $x$-value Of 0? Select Three Options.- $f(x)=|x|$- $f(x)=|x|+3$- $f(x)=|x+3|$- $f(x)=|x|-6$- $f(x)=|x+3|-6$
In mathematics, particularly in the study of functions, a vertex is a point where the function changes direction. It is a critical point that can be either a maximum or a minimum point on the graph of the function. When it comes to identifying functions with a vertex at x = 0, we need to consider the properties of absolute value functions, which are commonly used to model real-world phenomena.
Understanding Absolute Value Functions
Absolute value functions are defined as the distance of a number from zero on the number line. The general form of an absolute value function is f(x) = |x|, where x is the input variable. When x is positive, the absolute value function returns the value of x, and when x is negative, it returns the value of -x.
Analyzing the Given Functions
We are given five functions to analyze and determine which ones have a vertex with an x-value of 0:
To determine which functions have a vertex at x = 0, we need to examine the graph of each function and identify the point where the function changes direction.
Function 1: f(x) = |x|
The graph of f(x) = |x| is a V-shaped graph with its vertex at the origin (0, 0). This is because the absolute value function returns the distance of x from zero, which is always non-negative. Therefore, the vertex of this function is at x = 0.
Function 2: f(x) = |x| + 3
The graph of f(x) = |x| + 3 is a V-shaped graph with its vertex shifted 3 units upwards. This is because the absolute value function returns the distance of x from zero, and then we add 3 to it. Therefore, the vertex of this function is still at x = 0, but it is shifted 3 units upwards.
Function 3: f(x) = |x + 3|
The graph of f(x) = |x + 3| is a V-shaped graph with its vertex shifted 3 units to the left. This is because the absolute value function returns the distance of x + 3 from zero, which is always non-negative. Therefore, the vertex of this function is at x = -3, not x = 0.
Function 4: f(x) = |x| - 6
The graph of f(x) = |x| - 6 is a V-shaped graph with its vertex shifted 6 units downwards. This is because the absolute value function returns the distance of x from zero, and then we subtract 6 from it. Therefore, the vertex of this function is still at x = 0, but it is shifted 6 units downwards.
Function 5: f(x) = |x + 3| - 6
The graph of f(x) = |x + 3| - 6 is a V-shaped graph with its vertex shifted 3 units to the left and 6 units downwards. This is because the absolute value function returns the distance of x + 3 from zero, and then we subtract 6 from it. Therefore, the vertex of this function is at x = -3, not x = 0.
Conclusion
Based on our analysis, we can conclude that the functions with a vertex at x = 0 are:
These functions have a vertex at x = 0, which means that the function changes direction at this point. The other two functions, $f(x)=|x+3|$ and $f(x)=|x+3|-6$, have their vertices shifted to the left, and therefore do not have a vertex at x = 0.
Key Takeaways
- Absolute value functions are defined as the distance of a number from zero on the number line.
- The graph of an absolute value function is a V-shaped graph with its vertex at the origin (0, 0).
- Shifting the graph of an absolute value function up or down does not change the location of its vertex.
- Shifting the graph of an absolute value function to the left or right changes the location of its vertex.
In our previous article, we discussed the properties of absolute value functions and analyzed five functions to determine which ones have a vertex at x = 0. In this article, we will answer some frequently asked questions related to vertex functions.
Q: What is a vertex in a function?
A: A vertex is a point on the graph of a function where the function changes direction. It can be either a maximum or a minimum point on the graph.
Q: What is an absolute value function?
A: An absolute value function is a function that returns the distance of a number from zero on the number line. The general form of an absolute value function is f(x) = |x|, where x is the input variable.
Q: How do I identify the vertex of an absolute value function?
A: To identify the vertex of an absolute value function, you need to examine the graph of the function and find the point where the function changes direction. For an absolute value function, the vertex is always at the origin (0, 0).
Q: What happens when I shift the graph of an absolute value function up or down?
A: When you shift the graph of an absolute value function up or down, the vertex remains at the same location. The only thing that changes is the position of the graph relative to the x-axis.
Q: What happens when I shift the graph of an absolute value function to the left or right?
A: When you shift the graph of an absolute value function to the left or right, the vertex moves to a new location. The amount of shift is equal to the amount of shift in the x-direction.
Q: Can I have multiple vertices in a function?
A: Yes, it is possible to have multiple vertices in a function. However, in the case of absolute value functions, the vertex is always at the origin (0, 0).
Q: How do I determine which functions have a vertex at x = 0?
A: To determine which functions have a vertex at x = 0, you need to examine the graph of each function and identify the point where the function changes direction. For absolute value functions, the vertex is always at the origin (0, 0).
Q: What are some real-world applications of vertex functions?
A: Vertex functions have many real-world applications, such as modeling population growth, analyzing financial data, and understanding the behavior of physical systems.
Q: Can I use vertex functions to model real-world phenomena?
A: Yes, vertex functions can be used to model real-world phenomena. By understanding the properties of vertex functions, you can create mathematical models that accurately describe the behavior of complex systems.
Q: How do I use vertex functions to solve problems?
A: To use vertex functions to solve problems, you need to identify the vertex of the function and then use it to analyze the behavior of the system. This can involve shifting the graph of the function, analyzing the resulting graph, and drawing conclusions based on the analysis.
Conclusion
In this article, we answered some frequently asked questions related to vertex functions. We discussed the properties of absolute value functions, identified the vertex of each function, and analyzed the behavior of the functions. By understanding the properties of vertex functions, you can create mathematical models that accurately describe the behavior of complex systems and solve real-world problems.
Key Takeaways
- A vertex is a point on the graph of a function where the function changes direction.
- An absolute value function is a function that returns the distance of a number from zero on the number line.
- The vertex of an absolute value function is always at the origin (0, 0).
- Shifting the graph of an absolute value function up or down does not change the location of its vertex.
- Shifting the graph of an absolute value function to the left or right changes the location of its vertex.
- Vertex functions have many real-world applications, such as modeling population growth, analyzing financial data, and understanding the behavior of physical systems.