Analyzing the Completed Table

Oct 13, 2025 by windailys.com 56 views

Understanding the Value Function $y = |x| + 2$

The value function $y = |x| + 2$ is a mathematical expression that involves the absolute value of $x$ . The absolute value function $|x|$ returns the distance of $x$ from zero on the number line, without considering whether $x$ is positive or negative. When we add 2 to the absolute value of $x$ , we are essentially shifting the graph of the absolute value function up by 2 units.

Completing the Table for the Function $y = |x| + 2$

To complete the table for the function $y = |x| + 2$ , we need to evaluate the function for different values of $x$ . We can start by plugging in the given values of $x$ into the function and calculating the corresponding values of $y$ .

x	**y =	x	+ 2**
-2	4
-1	3
0	2
1	3
2	4

Let's start by evaluating the function for $x = -2$ . We have:

y = |x| + 2 = |-2| + 2 = 2 + 2 = 4

So, the value of $y$ for $x = -2$ is 4.

Next, let's evaluate the function for $x = -1$ . We have:

y = |x| + 2 = |-1| + 2 = 1 + 2 = 3

So, the value of $y$ for $x = -1$ is 3.

Now, let's evaluate the function for $x = 0$ . We have:

y = |x| + 2 = |0| + 2 = 0 + 2 = 2

So, the value of $y$ for $x = 0$ is 2.

Next, let's evaluate the function for $x = 1$ . We have:

y = |x| + 2 = |1| + 2 = 1 + 2 = 3

So, the value of $y$ for $x = 1$ is 3.

Finally, let's evaluate the function for $x = 2$ . We have:

y = |x| + 2 = |2| + 2 = 2 + 2 = 4

So, the value of $y$ for $x = 2$ is 4.

Analyzing the Completed Table

Now that we have completed the table for the function $y = |x| + 2$ , let's analyze the results.

We can see that the value of $y$ is always positive, regardless of the value of $x$ . This is because the absolute value function $|x|$ always returns a non-negative value, and adding 2 to it ensures that the result is always positive.

We can also see that the value of $y$ is symmetric about the y-axis. This means that for every value of $x$ to the left of the y-axis, there is a corresponding value of $x$ to the right of the y-axis that produces the same value of $y$ .

Graphing the Function

To visualize the function $y = |x| + 2$ , we can graph it on a coordinate plane.

The graph of the function $y = |x| + 2$ is a V-shaped graph that opens upwards. The vertex of the graph is at the point (0, 2), which is the minimum point of the graph.

The graph is symmetric about the y-axis, which means that for every point to the left of the y-axis, there is a corresponding point to the right of the y-axis that produces the same value of $y$ .

Conclusion

In conclusion, we have completed the table for the function $y = |x| + 2$ and analyzed the results. We have seen that the value of $y$ is always positive, regardless of the value of $x$ , and that the graph of the function is a V-shaped graph that opens upwards.

Key Takeaways

The value function $y = |x| + 2$ is a mathematical expression that involves the absolute value of $x$ .
The absolute value function $|x|$ returns the distance of $x$ from zero on the number line, without considering whether $x$ is positive or negative.
The graph of the function $y = |x| + 2$ is a V-shaped graph that opens upwards.
The vertex of the graph is at the point (0, 2), which is the minimum point of the graph.
The graph is symmetric about the y-axis.

Further Reading

If you want to learn more about value functions and absolute value functions, I recommend checking out the following resources:

Khan Academy: Absolute Value Functions
Math Is Fun: Absolute Value
Wolfram MathWorld: Absolute Value

I hope this article has helped you understand the value function $y = |x| + 2$ and its graph. If you have any questions or need further clarification, please don't hesitate to ask.
Q&A: Understanding the Value Function $y = |x| + 2$

In our previous article, we explored the value function $y = |x| + 2$ and its graph. We saw that the value of $y$ is always positive, regardless of the value of $x$ , and that the graph of the function is a V-shaped graph that opens upwards. In this article, we will answer some frequently asked questions about the value function $y = |x| + 2$ .

Q: What is the absolute value function?

A: The absolute value function $|x|$ returns the distance of $x$ from zero on the number line, without considering whether $x$ is positive or negative. For example, $|3| = 3$ and $|-3| = 3$ .

Q: How does the absolute value function relate to the value function $y = |x| + 2$ ?

A: The absolute value function $|x|$ is the core component of the value function $y = |x| + 2$ . When we add 2 to the absolute value of $x$ , we are essentially shifting the graph of the absolute value function up by 2 units.

Q: What is the vertex of the graph of the value function $y = |x| + 2$ ?

A: The vertex of the graph of the value function $y = |x| + 2$ is at the point (0, 2), which is the minimum point of the graph.

Q: Is the graph of the value function $y = |x| + 2$ symmetric about the y-axis?

A: Yes, the graph of the value function $y = |x| + 2$ is symmetric about the y-axis. This means that for every point to the left of the y-axis, there is a corresponding point to the right of the y-axis that produces the same value of $y$ .

Q: Can the value function $y = |x| + 2$ be used to model real-world phenomena?

A: Yes, the value function $y = |x| + 2$ can be used to model real-world phenomena that involve absolute values, such as the distance between two points or the magnitude of a vector.

Q: How can I graph the value function $y = |x| + 2$ ?

A: To graph the value function $y = |x| + 2$ , you can use a coordinate plane and plot the points (x, y) for different values of x. You can also use a graphing calculator or software to visualize the graph.

Q: What are some common applications of the value function $y = |x| + 2$ ?

A: Some common applications of the value function $y = |x| + 2$ include:

Modeling the distance between two points
Modeling the magnitude of a vector
Modeling the absolute value of a quantity
Modeling the minimum or maximum value of a function

Q: Can I use the value function $y = |x| + 2$ to solve equations or inequalities?

A: Yes, you can use the value function $y = |x| + 2$ to solve equations or inequalities that involve absolute values. For example, you can use the value function to solve equations of the form $|x| = a$ or $|x| > a$ .

Q: How can I use the value function $y = |x| + 2$ to model real-world phenomena?

A: To use the value function $y = |x| + 2$ to model real-world phenomena, you can start by identifying the key variables and relationships involved in the phenomenon. You can then use the value function to model the absolute value of the variables or the relationships between them.

Conclusion

In conclusion, the value function $y = |x| + 2$ is a powerful tool for modeling real-world phenomena that involve absolute values. By understanding the properties and applications of the value function, you can use it to solve equations and inequalities, model real-world phenomena, and visualize the graph of the function.

Key Takeaways

The absolute value function $|x|$ returns the distance of $x$ from zero on the number line, without considering whether $x$ is positive or negative.
The value function $y = |x| + 2$ is a V-shaped graph that opens upwards, with the vertex at the point (0, 2).
The graph of the value function $y = |x| + 2$ is symmetric about the y-axis.
The value function $y = |x| + 2$ can be used to model real-world phenomena that involve absolute values.
The value function $y = |x| + 2$ can be used to solve equations and inequalities that involve absolute values.

Further Reading

If you want to learn more about the value function $y = |x| + 2$ and its applications, I recommend checking out the following resources:

Khan Academy: Absolute Value Functions
Math Is Fun: Absolute Value
Wolfram MathWorld: Absolute Value
MIT OpenCourseWare: Calculus II
Wolfram Alpha: Value Function