Which Could Be The Graph Of F ( X ) = ∣ X − H ∣ + K F(x)=|x-h|+k F ( X ) = ∣ X − H ∣ + K If H H H And K K K Are Both Positive?

Introduction

When it comes to graphing functions, understanding the properties and behavior of different types of functions is crucial. In this article, we will delve into the graph of the function $f(x)=|x-h|+k$, where $h$ and $k$ are both positive. This function is a type of absolute value function, which is a fundamental concept in mathematics.

Understanding Absolute Value Functions

Absolute value functions are a type of function that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, it is the magnitude of the number. The absolute value function is denoted by the symbol $|x|$ and is defined as:

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

Graphing Absolute Value Functions

The graph of an absolute value function is a V-shaped graph that opens upwards or downwards. The vertex of the V is at the point where the function changes from increasing to decreasing or vice versa. For the function $f(x)=|x-h|+k$, the vertex will be at the point $(h, k)$.

Properties of the Function $f(x)=|x-h|+k$

Since $h$ and $k$ are both positive, the function $f(x)=|x-h|+k$ will have the following properties:

• The vertex of the V-shaped graph will be at the point $(h, k)$.
• The graph will open upwards.
• The function will be increasing for $x > h$ and decreasing for $x < h$.
• The function will have a minimum value at the vertex, which is $k$.

Graphing the Function $f(x)=|x-h|+k$

To graph the function $f(x)=|x-h|+k$, we can follow these steps:

1. Plot the vertex at the point $(h, k)$.
2. Draw a V-shaped graph that opens upwards, with the vertex at the point $(h, k)$.
3. The graph will be increasing for $x > h$ and decreasing for $x < h$.
4. The function will have a minimum value at the vertex, which is $k$.

Examples of Graphs of $f(x)=|x-h|+k$

Here are some examples of graphs of the function $f(x)=|x-h|+k$ for different values of $h$ and $k$:

• If $h=2$ and $k=3$, the graph of $f(x)=|x-2|+3$ will be a V-shaped graph that opens upwards, with the vertex at the point $(2, 3)$.
• If $h=4$ and $k=5$, the graph of $f(x)=|x-4|+5$ will be a V-shaped graph that opens upwards, with the vertex at the point $(4, 5)$.
• If $h=1$ and $k=2$, the graph of $f(x)=|x-1|+2$ will be a V-shaped graph that opens upwards, with the vertex at the point $(1, 2)$.

Conclusion

In conclusion, the graph of the function $f(x)=|x-h|+k$ is a V-shaped graph that opens upwards, with the vertex at the point $(h, k)$. The function has a minimum value at the vertex, which is $k$. The graph will be increasing for $x > h$ and decreasing for $x < h$. Understanding the properties and behavior of absolute value functions is crucial in graphing and analyzing functions in mathematics.

References

• [1] "Absolute Value Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/absolutevalue.html
• [2] "Graphing Absolute Value Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphabs.htm
• [3] "Properties of Absolute Value Functions" by Mathway. Retrieved from https://www.mathway.com/subjects/absolute-value/functions/properties

Further Reading

• "Graphing Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5b7c7c:graphing-functions
• "Absolute Value Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/AbsoluteValueFunction.html
  

Introduction

In our previous article, we discussed the graph of the function $f(x)=|x-h|+k$, where $h$ and $k$ are both positive. We explored the properties and behavior of this function, including its V-shaped graph and minimum value at the vertex. In this article, we will answer some frequently asked questions about graphing the function $f(x)=|x-h|+k$.

Q: What is the vertex of the graph of $f(x)=|x-h|+k$?

A: The vertex of the graph of $f(x)=|x-h|+k$ is the point $(h, k)$.

Q: How do I graph the function $f(x)=|x-h|+k$?

A: To graph the function $f(x)=|x-h|+k$, follow these steps:

1. Plot the vertex at the point $(h, k)$.
2. Draw a V-shaped graph that opens upwards, with the vertex at the point $(h, k)$.
3. The graph will be increasing for $x > h$ and decreasing for $x < h$.
4. The function will have a minimum value at the vertex, which is $k$.

Q: What is the minimum value of the function $f(x)=|x-h|+k$?

A: The minimum value of the function $f(x)=|x-h|+k$ is $k$, which occurs at the vertex $(h, k)$.

Q: How do I determine the value of $h$ and $k$ for a given graph of $f(x)=|x-h|+k$?

A: To determine the value of $h$ and $k$ for a given graph of $f(x)=|x-h|+k$, follow these steps:

1. Identify the vertex of the graph, which is the point $(h, k)$.
2. The value of $h$ is the x-coordinate of the vertex.
3. The value of $k$ is the y-coordinate of the vertex.

Q: Can the graph of $f(x)=|x-h|+k$ be a horizontal line?

A: No, the graph of $f(x)=|x-h|+k$ cannot be a horizontal line. The function is a V-shaped graph that opens upwards, with the vertex at the point $(h, k)$.

Q: Can the graph of $f(x)=|x-h|+k$ be a vertical line?

A: No, the graph of $f(x)=|x-h|+k$ cannot be a vertical line. The function is a V-shaped graph that opens upwards, with the vertex at the point $(h, k)$.

Q: How do I graph the function $f(x)=|x-h|+k$ using a graphing calculator?

A: To graph the function $f(x)=|x-h|+k$ using a graphing calculator, follow these steps:

1. Enter the function $f(x)=|x-h|+k$ into the calculator.
2. Set the x-axis to the correct scale.
3. Plot the graph of the function.
4. Identify the vertex of the graph, which is the point $(h, k)$.

Q: Can I graph the function $f(x)=|x-h|+k$ using a computer algebra system (CAS)?

A: Yes, you can graph the function $f(x)=|x-h|+k$ using a computer algebra system (CAS). Follow the same steps as graphing the function using a graphing calculator.

Conclusion

In conclusion, graphing the function $f(x)=|x-h|+k$ requires understanding the properties and behavior of absolute value functions. By following the steps outlined in this article, you can graph the function and identify its vertex, minimum value, and other key features.

References

• [1] "Graphing Absolute Value Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphabs.htm
• [2] "Properties of Absolute Value Functions" by Mathway. Retrieved from https://www.mathway.com/subjects/absolute-value/functions/properties
• [3] "Graphing Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f5b7c7c:graphing-functions

Further Reading

• "Absolute Value Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/AbsoluteValueFunction.html
• "Graphing Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/graphing.html