Select The Correct Answer.What Is The Vertex Of The Function $f(x)=|x-9|+2$?A. $(9, 2)$ B. $ ( − 9 , − 2 ) (-9, -2) ( − 9 , − 2 ) [/tex] C. $(-9, 2)$ D. $(9, -2)$

When dealing with functions, particularly those involving absolute values, understanding the concept of a vertex is crucial. The vertex of a function is the point at which the function changes direction, and it is a critical point in the graph of the function. In this article, we will explore the concept of a vertex and how to find it for a given function.

What is a Vertex?

A vertex is a point on the graph of a function where the function changes direction. It is the point at which the function has a minimum or maximum value. For a function that opens upwards or downwards, the vertex is the lowest or highest point on the graph, respectively.

Finding the Vertex of a Function

To find the vertex of a function, we need to analyze the function and determine its type. If the function is a quadratic function, we can use the formula for the vertex:

$$x = -\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the quadratic function.

However, if the function involves absolute values, we need to use a different approach. In this case, we can rewrite the function as a piecewise function and then find the vertex.

Rewriting the Function

The given function is:

$$f(x) = |x-9|+2$$

We can rewrite this function as a piecewise function:

$$f(x) = \begin{cases} x-9+2 & \text{if } x \geq 9 \\ -(x-9)+2 & \text{if } x < 9 \end{cases}$$

Finding the Vertex

To find the vertex, we need to analyze the two cases separately.

For the case $x \geq 9$, the function is:

$$f(x) = x-9+2 = x-7$$

This is a linear function, and its vertex is at the point where the function changes direction. Since the function is increasing for $x \geq 9$, the vertex is at the point where $x = 9$.

For the case $x < 9$, the function is:

$$f(x) = -(x-9)+2 = -x+11$$

This is also a linear function, and its vertex is at the point where the function changes direction. Since the function is decreasing for $x < 9$, the vertex is at the point where $x = 9$.

Conclusion

In conclusion, the vertex of the function $f(x) = |x-9|+2$ is at the point $(9, 2)$.

Answer

The correct answer is:

A. $(9, 2)$

Discussion

The vertex of a function is an important concept in mathematics, particularly in calculus and graph theory. Understanding the concept of a vertex is crucial for analyzing and graphing functions. In this article, we explored the concept of a vertex and how to find it for a given function. We also analyzed the given function and determined its vertex.

Related Topics

• Quadratic Functions: Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$.
• Absolute Value Functions: Absolute value functions are a type of function that involves the absolute value of a variable. They can be written in the form $f(x) = |x-a|+b$. The vertex of an absolute value function can be found by rewriting the function as a piecewise function and then analyzing the two cases separately.
• Graph Theory: Graph theory is a branch of mathematics that deals with the study of graphs, which are collections of vertices and edges. Understanding the concept of a vertex is crucial for graph theory, particularly for analyzing and graphing functions.

References

• Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. Understanding the concept of a vertex is crucial for calculus, particularly for analyzing and graphing functions.
• Graph Theory: Graph theory is a branch of mathematics that deals with the study of graphs, which are collections of vertices and edges. Understanding the concept of a vertex is crucial for graph theory, particularly for analyzing and graphing functions.

Conclusion

In the previous article, we explored the concept of a vertex and how to find it for a given function. In this article, we will answer some frequently asked questions about the vertex of a function.

Q: What is the vertex of a function?

A: The vertex of a function is the point at which the function changes direction. It is the point at which the function has a minimum or maximum value.

Q: How do I find the vertex of a function?

A: To find the vertex of a function, you need to analyze the function and determine its type. If the function is a quadratic function, you can use the formula for the vertex:

$$x = -\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the quadratic function.

However, if the function involves absolute values, you need to use a different approach. In this case, you can rewrite the function as a piecewise function and then find the vertex.

Q: What is the difference between the vertex and the minimum or maximum value of a function?

A: The vertex of a function is the point at which the function changes direction, whereas the minimum or maximum value of a function is the lowest or highest point on the graph of the function.

Q: Can the vertex of a function be a minimum or maximum value?

A: Yes, the vertex of a function can be a minimum or maximum value. For example, if the function is a quadratic function that opens upwards, the vertex is the minimum value of the function.

Q: How do I determine the type of a function?

A: To determine the type of a function, you need to analyze the function and look for the following characteristics:

• Quadratic function: A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$.
• Absolute value function: An absolute value function is a function that involves the absolute value of a variable. It can be written in the form $f(x) = |x-a|+b$. The vertex of an absolute value function can be found by rewriting the function as a piecewise function and then analyzing the two cases separately.

Q: Can the vertex of a function be a point of inflection?

A: Yes, the vertex of a function can be a point of inflection. A point of inflection is a point on the graph of a function where the function changes from concave up to concave down or vice versa.

Q: How do I find the point of inflection of a function?

A: To find the point of inflection of a function, you need to analyze the function and look for the following characteristics:

• Concave up: A function is concave up if its second derivative is positive.
• Concave down: A function is concave down if its second derivative is negative.

You can use the following formula to find the point of inflection of a function:

$$x = -\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the second derivative of the function.

Q: Can the vertex of a function be a point of discontinuity?

A: Yes, the vertex of a function can be a point of discontinuity. A point of discontinuity is a point on the graph of a function where the function is not defined.

Q: How do I find the point of discontinuity of a function?

A: To find the point of discontinuity of a function, you need to analyze the function and look for the following characteristics:

• Undefined: A function is undefined if it is not defined at a particular point.
• Discontinuous: A function is discontinuous if it has a gap or a jump at a particular point.

You can use the following formula to find the point of discontinuity of a function:

$$x = \frac{a}{b}$$

where $a$ and $b$ are the coefficients of the function.

Conclusion

In conclusion, the vertex of a function is an important concept in mathematics, particularly in calculus and graph theory. Understanding the concept of a vertex is crucial for analyzing and graphing functions. In this article, we answered some frequently asked questions about the vertex of a function and provided examples and formulas to help you understand the concept.

Related Topics

• Quadratic Functions: Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$.
• Absolute Value Functions: Absolute value functions are a type of function that involves the absolute value of a variable. They can be written in the form $f(x) = |x-a|+b$. The vertex of an absolute value function can be found by rewriting the function as a piecewise function and then analyzing the two cases separately.
• Graph Theory: Graph theory is a branch of mathematics that deals with the study of graphs, which are collections of vertices and edges. Understanding the concept of a vertex is crucial for graph theory, particularly for analyzing and graphing functions.

References

• Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. Understanding the concept of a vertex is crucial for calculus, particularly for analyzing and graphing functions.
• Graph Theory: Graph theory is a branch of mathematics that deals with the study of graphs, which are collections of vertices and edges. Understanding the concept of a vertex is crucial for graph theory, particularly for analyzing and graphing functions.