Understanding the Basics of Absolute Value Functions

Oct 13, 2025 by leolavanderia.com.br 131 views

Understanding the Basics of Absolute Value Functions

Absolute value functions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. The graph of y = |x| is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). This graph represents the absolute value of x, which means that for any value of x, the corresponding value of y is the distance of x from 0, without considering the direction.

Transforming the Graph of y = |x|

To find the graph of f(x) = |x - 9|, we need to understand how the graph of y = |x| can be transformed. The function f(x) = |x - 9| is a transformation of the graph of y = |x|, where the graph has been shifted 9 units to the right. This means that for any point (x, y) on the graph of y = |x|, the corresponding point on the graph of f(x) = |x - 9| is (x + 9, y).

Understanding the Concept of Horizontal Shift

A horizontal shift is a transformation that moves the graph of a function to the left or right. In this case, the graph of y = |x| has been shifted 9 units to the right to obtain the graph of f(x) = |x - 9|. This means that for any point (x, y) on the graph of y = |x|, the corresponding point on the graph of f(x) = |x - 9| is (x + 9, y).

Visualizing the Graph of f(x) = |x - 9|

To visualize the graph of f(x) = |x - 9|, we can start with the graph of y = |x| and then shift it 9 units to the right. This will result in a V-shaped graph that opens upwards, with its vertex at the point (9, 0). The graph will have the same shape as the graph of y = |x|, but it will be shifted 9 units to the right.

Key Features of the Graph of f(x) = |x - 9|

The graph of f(x) = |x - 9| has several key features that are worth noting. The vertex of the graph is at the point (9, 0), and the graph opens upwards. The graph is symmetric about the vertical line x = 9, and it has a minimum value of 0 at x = 9. The graph also has a maximum value of 9 at x = 18 and x = -18.

Finding the Graph of f(x) = |x - 9| Using Transformations

To find the graph of f(x) = |x - 9| using transformations, we can start with the graph of y = |x| and then apply the following transformations:

Shift the graph 9 units to the right.

This will result in a V-shaped graph that opens upwards, with its vertex at the point (9, 0). The graph will have the same shape as the graph of y = |x|, but it will be shifted 9 units to the right.

Conclusion

In conclusion, the graph of f(x) = |x - 9| can be found by using the graph of y = |x| and applying a horizontal shift of 9 units to the right. This transformation results in a V-shaped graph that opens upwards, with its vertex at the point (9, 0). The graph is symmetric about the vertical line x = 9, and it has a minimum value of 0 at x = 9. The graph also has a maximum value of 9 at x = 18 and x = -18.

Key Takeaways

The graph of f(x) = |x - 9| is a transformation of the graph of y = |x|, where the graph has been shifted 9 units to the right.
The graph of f(x) = |x - 9| has a vertex at the point (9, 0) and opens upwards.
The graph is symmetric about the vertical line x = 9.
The graph has a minimum value of 0 at x = 9 and a maximum value of 9 at x = 18 and x = -18.

Real-World Applications

The concept of absolute value functions and their transformations has numerous real-world applications. For example, in physics, the absolute value function can be used to model the motion of an object, where the absolute value of the object's velocity represents the distance traveled by the object. In engineering, the absolute value function can be used to model the behavior of electrical circuits, where the absolute value of the voltage represents the magnitude of the current flowing through the circuit.

Practice Problems

Find the graph of f(x) = |x + 3| by using the graph of y = |x|.
Find the graph of f(x) = |x - 2| by using the graph of y = |x|.
Find the graph of f(x) = |x + 5| by using the graph of y = |x|.

Solutions

To find the graph of f(x) = |x + 3|, we can start with the graph of y = |x| and then shift it 3 units to the left. This will result in a V-shaped graph that opens upwards, with its vertex at the point (-3, 0).
To find the graph of f(x) = |x - 2|, we can start with the graph of y = |x| and then shift it 2 units to the right. This will result in a V-shaped graph that opens upwards, with its vertex at the point (2, 0).
To find the graph of f(x) = |x + 5|, we can start with the graph of y = |x| and then shift it 5 units to the left. This will result in a V-shaped graph that opens upwards, with its vertex at the point (-5, 0).

Conclusion

Q: What is the graph of y = |x|?

A: The graph of y = |x| is a V-shaped graph that opens upwards, with its vertex at the origin (0, 0). This graph represents the absolute value of x, which means that for any value of x, the corresponding value of y is the distance of x from 0, without considering the direction.

Q: How can I find the graph of f(x) = |x - 9| using the graph of y = |x|?

A: To find the graph of f(x) = |x - 9| using the graph of y = |x|, you can apply a horizontal shift of 9 units to the right. This means that for any point (x, y) on the graph of y = |x|, the corresponding point on the graph of f(x) = |x - 9| is (x + 9, y).

Q: What is the vertex of the graph of f(x) = |x - 9|?

A: The vertex of the graph of f(x) = |x - 9| is at the point (9, 0).

Q: Is the graph of f(x) = |x - 9| symmetric about any line?

A: Yes, the graph of f(x) = |x - 9| is symmetric about the vertical line x = 9.

Q: What are the minimum and maximum values of the graph of f(x) = |x - 9|?

A: The minimum value of the graph of f(x) = |x - 9| is 0 at x = 9, and the maximum value is 9 at x = 18 and x = -18.

Q: How can I use the graph of f(x) = |x - 9| in real-world applications?

A: The graph of f(x) = |x - 9| can be used to model various real-world phenomena, such as the motion of an object, the behavior of electrical circuits, and the growth of populations.

Q: Can I find the graph of f(x) = |x + 3| using the graph of y = |x|?

A: Yes, you can find the graph of f(x) = |x + 3| by applying a horizontal shift of 3 units to the left to the graph of y = |x|.

Q: Can I find the graph of f(x) = |x - 2| using the graph of y = |x|?

A: Yes, you can find the graph of f(x) = |x - 2| by applying a horizontal shift of 2 units to the right to the graph of y = |x|.

Q: Can I find the graph of f(x) = |x + 5| using the graph of y = |x|?

A: Yes, you can find the graph of f(x) = |x + 5| by applying a horizontal shift of 5 units to the left to the graph of y = |x|.

Q: What are some common transformations that can be applied to the graph of y = |x|?

A: Some common transformations that can be applied to the graph of y = |x| include horizontal shifts, vertical shifts, and reflections.

Q: How can I apply a horizontal shift to the graph of y = |x|?

A: To apply a horizontal shift to the graph of y = |x|, you can replace x with (x + h) or (x - h), where h is the amount of the shift.

Q: How can I apply a vertical shift to the graph of y = |x|?

A: To apply a vertical shift to the graph of y = |x|, you can multiply y by a constant factor or add a constant value to y.

Q: How can I apply a reflection to the graph of y = |x|?

A: To apply a reflection to the graph of y = |x|, you can replace x with (-x) or replace y with (-y).