What Is The Vertex Of The Absolute Value Function Defined By $f(x) = |x-7| + 1$?A. (7, 1) B. (-7, -1) C. (7, -1) D. (-7, 1)

Feb 28, 2025 by ADMIN 129 views

**Understanding the Absolute Value Function**

The absolute value function is a fundamental concept in mathematics, and it plays a crucial role in various mathematical disciplines, including algebra, geometry, and calculus. In this article, we will delve into the concept of the absolute value function and explore the vertex of the absolute value function defined by $f(x) = |x-7| + 1$ .

What is the Absolute Value Function?

The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. It is denoted by the symbol $|x|$ and is defined as:

|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

In other words, the absolute value function takes a number as input and returns its distance from zero. For example, if we input $x = 3$ , the absolute value function returns $|3| = 3$ , which is the distance of $3$ from zero on the number line.

The Absolute Value Function Defined by $f(x) = |x-7| + 1$

The absolute value function defined by $f(x) = |x-7| + 1$ is a specific type of absolute value function that has been shifted to the right by $7$ units and up by $1$ unit. This means that the graph of the function is a vertical shift of the graph of the standard absolute value function by $7$ units to the right and $1$ unit up.

Finding the Vertex of the Absolute Value Function

The vertex of a function is the point on the graph of the function where the function reaches its minimum or maximum value. In the case of the absolute value function defined by $f(x) = |x-7| + 1$ , the vertex is the point where the function reaches its minimum value.

To find the vertex of the absolute value function, we need to find the value of $x$ that minimizes the function. Since the absolute value function is a piecewise function, we need to consider two cases: when $x \geq 7$ and when $x < 7$ .

Case 1: $x \geq 7$

When $x \geq 7$ , the absolute value function is defined as $f(x) = x-7+1 = x-6$ . To find the minimum value of the function, we need to find the value of $x$ that minimizes the expression $x-6$ . Since the expression is a linear function, the minimum value occurs when $x$ is equal to the x-intercept of the line, which is $x = 6$ . However, since $x \geq 7$ , the minimum value of the function occurs when $x = 7$ .

Case 2: $x < 7$

When $x < 7$ , the absolute value function is defined as $f(x) = -(x-7)+1 = -x+8$ . To find the minimum value of the function, we need to find the value of $x$ that minimizes the expression $-x+8$ . Since the expression is a linear function, the minimum value occurs when $x$ is equal to the x-intercept of the line, which is $x = 8$ . However, since $x < 7$ , the minimum value of the function occurs when $x = 7$ .

Conclusion

In conclusion, the vertex of the absolute value function defined by $f(x) = |x-7| + 1$ is the point $(7, 1)$ . This is because the function reaches its minimum value when $x = 7$ , and the minimum value of the function is $1$ .

Answer

The correct answer is A. (7, 1).

Additional Tips and Tricks

The absolute value function is a fundamental concept in mathematics, and it plays a crucial role in various mathematical disciplines, including algebra, geometry, and calculus.
The absolute value function defined by $f(x) = |x-7| + 1$ is a specific type of absolute value function that has been shifted to the right by $7$ units and up by $1$ unit.
The vertex of a function is the point on the graph of the function where the function reaches its minimum or maximum value.
To find the vertex of the absolute value function, we need to find the value of $x$ that minimizes the function.
The absolute value function is a piecewise function, and we need to consider two cases: when $x \geq 7$ and when $x < 7$ .

References

[1] "Absolute Value Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
[2] "Vertex of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola.htm

Frequently Asked Questions

What is the absolute value function?
What is the vertex of the absolute value function defined by $f(x) = |x-7| + 1$ ?
How do I find the vertex of the absolute value function?
What are some additional tips and tricks for working with absolute value functions?
Absolute Value Function Q&A =============================

Q: What is the absolute value function?

A: The absolute value function is a mathematical function that returns the distance of a number from zero on the number line. It is denoted by the symbol $|x|$ and is defined as:

|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}

Q: What is the vertex of the absolute value function defined by $f(x) = |x-7| + 1$ ?

A: The vertex of the absolute value function defined by $f(x) = |x-7| + 1$ is the point $(7, 1)$ . This is because the function reaches its minimum value when $x = 7$ , and the minimum value of the function is $1$ .

Q: How do I find the vertex of the absolute value function?

A: To find the vertex of the absolute value function, we need to find the value of $x$ that minimizes the function. Since the absolute value function is a piecewise function, we need to consider two cases: when $x \geq 7$ and when $x < 7$ .

Q: What are some additional tips and tricks for working with absolute value functions?

A: Here are some additional tips and tricks for working with absolute value functions:

The absolute value function is a fundamental concept in mathematics, and it plays a crucial role in various mathematical disciplines, including algebra, geometry, and calculus.
The absolute value function defined by $f(x) = |x-7| + 1$ is a specific type of absolute value function that has been shifted to the right by $7$ units and up by $1$ unit.
The vertex of a function is the point on the graph of the function where the function reaches its minimum or maximum value.
To find the vertex of the absolute value function, we need to find the value of $x$ that minimizes the function.
The absolute value function is a piecewise function, and we need to consider two cases: when $x \geq 7$ and when $x < 7$ .

Q: How do I graph the absolute value function?

A: To graph the absolute value function, we need to consider two cases: when $x \geq 7$ and when $x < 7$ . When $x \geq 7$ , the absolute value function is defined as $f(x) = x-7+1 = x-6$ . When $x < 7$ , the absolute value function is defined as $f(x) = -(x-7)+1 = -x+8$ .

Q: What are some common mistakes to avoid when working with absolute value functions?

A: Here are some common mistakes to avoid when working with absolute value functions:

Not considering the two cases when $x \geq 7$ and when $x < 7$ .
Not using the correct definition of the absolute value function.
Not graphing the absolute value function correctly.
Not finding the vertex of the absolute value function correctly.

Q: How do I use the absolute value function in real-world applications?

A: The absolute value function has many real-world applications, including:

Modeling the distance between two points on a number line.
Modeling the magnitude of a vector.
Modeling the absolute temperature.
Modeling the absolute humidity.

Q: What are some advanced topics related to absolute value functions?

A: Here are some advanced topics related to absolute value functions:

Absolute value inequalities.
Absolute value equations.
Absolute value functions with multiple variables.
Absolute value functions with complex numbers.

Q: How do I learn more about absolute value functions?

A: Here are some resources to learn more about absolute value functions:

Online tutorials and videos.
Textbooks and workbooks.
Online courses and lectures.
Practice problems and exercises.

Q: What are some common misconceptions about absolute value functions?

A: Here are some common misconceptions about absolute value functions:

The absolute value function is only used in mathematics.
The absolute value function is only used in algebra.
The absolute value function is only used in geometry.
The absolute value function is only used in calculus.

Q: How do I use technology to help me with absolute value functions?

A: Here are some ways to use technology to help you with absolute value functions:

Graphing calculators.
Computer algebra systems.
Online graphing tools.
Online calculators.

Q: What are some real-world examples of absolute value functions?

A: Here are some real-world examples of absolute value functions:

Modeling the distance between two points on a number line.
Modeling the magnitude of a vector.
Modeling the absolute temperature.
Modeling the absolute humidity.

Q: How do I apply absolute value functions to solve real-world problems?

A: Here are some steps to apply absolute value functions to solve real-world problems:

Identify the problem and the variables involved.
Determine the type of absolute value function needed.
Use the absolute value function to model the problem.
Solve the equation or inequality.
Interpret the results in the context of the problem.