Which Could Be The Graph Of $f(x) = |x - H| + K$ If $ H H H [/tex] And $k$ Are Both Positive?

Mar 12, 2025 by ADMIN 98 views

**Understanding the Graph of a Function with Absolute Value**

===========================================================

Introduction

In mathematics, the absolute value function is a fundamental concept that plays a crucial role in various mathematical operations. The graph of a function with absolute value can be complex and may have multiple forms depending on the values of the parameters involved. In this article, we will explore the graph of the function $f(x) = |x - h| + k$, where $h$ and $k$ are both positive.

The Absolute Value Function

The absolute value function is defined as $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$. This function takes the distance of a number from zero on the number line and returns that distance without considering whether it's positive or negative. The graph of the absolute value function is a V-shaped graph with its vertex at the origin.

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

The function $f(x) = |x - h| + k$ is a transformation of the absolute value function. The parameter $h$ represents a horizontal shift, and the parameter $k$ represents a vertical shift. When $h$ and $k$ are both positive, the graph of the function will be a V-shaped graph with its vertex at $(h, k)$.

Graphing the Function

To graph the function $f(x) = |x - h| + k$, we need to consider two cases: when $x \geq h$ and when $x < h$.

Case 1: $x \geq h$

When $x \geq h$, the absolute value function becomes $|x - h| = x - h$. Therefore, the function $f(x) = |x - h| + k$ becomes $f(x) = x - h + k = x + (k - h)$. This is a linear function with a slope of 1 and a y-intercept of $(k - h)$.

Case 2: $x < h$

When $x < h$, the absolute value function becomes $|x - h| = -(x - h) = h - x$. Therefore, the function $f(x) = |x - h| + k$ becomes $f(x) = h - x + k = -x + (h + k)$. This is also a linear function with a slope of -1 and a y-intercept of $(h + k)$.

The Graph of the Function

Based on the two cases, we can conclude that the graph of the function $f(x) = |x - h| + k$ is a V-shaped graph with its vertex at $(h, k)$. The graph consists of two linear segments: one with a slope of 1 and a y-intercept of $(k - h)$, and the other with a slope of -1 and a y-intercept of $(h + k)$.

Conclusion

In conclusion, the graph of the function $f(x) = |x - h| + k$, where $h$ and $k$ are both positive, is a V-shaped graph with its vertex at $(h, k)$. The graph consists of two linear segments with slopes of 1 and -1, respectively, and y-intercepts of $(k - h)$ and $(h + k)$, respectively. This graph is a fundamental concept in mathematics and has numerous applications in various fields.

Example Problems

Problem 1

Graph the function $f(x) = |x - 2| + 3$.

Solution

To graph the function $f(x) = |x - 2| + 3$, we need to consider two cases: when $x \geq 2$ and when $x < 2$.

When $x \geq 2$, the absolute value function becomes $|x - 2| = x - 2$. Therefore, the function $f(x) = |x - 2| + 3$ becomes $f(x) = x - 2 + 3 = x + 1$. This is a linear function with a slope of 1 and a y-intercept of 1.

When $x < 2$, the absolute value function becomes $|x - 2| = -(x - 2) = 2 - x$. Therefore, the function $f(x) = |x - 2| + 3$ becomes $f(x) = 2 - x + 3 = -x + 5$. This is also a linear function with a slope of -1 and a y-intercept of 5.

The graph of the function $f(x) = |x - 2| + 3$ is a V-shaped graph with its vertex at $(2, 3)$. The graph consists of two linear segments: one with a slope of 1 and a y-intercept of 1, and the other with a slope of -1 and a y-intercept of 5.

Problem 2

Graph the function $f(x) = |x - 4| + 2$.

Solution

To graph the function $f(x) = |x - 4| + 2$, we need to consider two cases: when $x \geq 4$ and when $x < 4$.

When $x \geq 4$, the absolute value function becomes $|x - 4| = x - 4$. Therefore, the function $f(x) = |x - 4| + 2$ becomes $f(x) = x - 4 + 2 = x - 2$. This is a linear function with a slope of 1 and a y-intercept of -2.

When $x < 4$, the absolute value function becomes $|x - 4| = -(x - 4) = 4 - x$. Therefore, the function $f(x) = |x - 4| + 2$ becomes $f(x) = 4 - x + 2 = -x + 6$. This is also a linear function with a slope of -1 and a y-intercept of 6.

The graph of the function $f(x) = |x - 4| + 2$ is a V-shaped graph with its vertex at $(4, 2)$. The graph consists of two linear segments: one with a slope of 1 and a y-intercept of -2, and the other with a slope of -1 and a y-intercept of 6.

Final Thoughts

=====================================================================================

Q: What is the graph of a function with absolute value?

A: The graph of a function with absolute value is a V-shaped graph that consists of two linear segments. The graph has a vertex at the point where the absolute value function changes its behavior.

Q: What are the two cases for graphing a function with absolute value?

A: There are two cases for graphing a function with absolute value:

When $x \geq h$, the absolute value function becomes $|x - h| = x - h$.
When $x < h$, the absolute value function becomes $|x - h| = -(x - h) = h - x$.

Q: How do I determine the slope and y-intercept of the two linear segments?

A: To determine the slope and y-intercept of the two linear segments, you need to consider the two cases:

When $x \geq h$, the slope is 1 and the y-intercept is $(k - h)$.
When $x < h$, the slope is -1 and the y-intercept is $(h + k)$.

Q: What is the vertex of the graph of a function with absolute value?

A: The vertex of the graph of a function with absolute value is the point where the absolute value function changes its behavior. The vertex is located at the point $(h, k)$.

Q: How do I graph a function with absolute value?

A: To graph a function with absolute value, you need to follow these steps:

Determine the values of $h$ and $k$.
Graph the two linear segments using the slopes and y-intercepts.
Connect the two linear segments at the vertex.

Q: What are some examples of functions with absolute value?

A: Some examples of functions with absolute value include:

$f(x) = |x - 2| + 3$
$f(x) = |x - 4| + 2$
$f(x) = |x + 1| - 2$

Q: How do I find the equation of a function with absolute value?

A: To find the equation of a function with absolute value, you need to follow these steps:

Determine the values of $h$ and $k$.
Write the equation in the form $f(x) = |x - h| + k$.
Simplify the equation to find the final form.

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

A: Some real-world applications of functions with absolute value include:

Modeling the distance between two points.
Modeling the temperature difference between two locations.
Modeling the cost of a product based on its weight.

Q: How do I use functions with absolute value in real-world problems?

A: To use functions with absolute value in real-world problems, you need to follow these steps:

Identify the problem and determine the values of $h$ and $k$.
Write the equation in the form $f(x) = |x - h| + k$.
Solve the equation to find the final answer.

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

A: Some common mistakes to avoid when graphing functions with absolute value include:

Not considering the two cases for graphing.
Not determining the slope and y-intercept correctly.
Not connecting the two linear segments at the vertex.

Q: How do I check my work when graphing functions with absolute value?

A: To check your work when graphing functions with absolute value, you need to follow these steps:

Verify that the two linear segments are connected at the vertex.
Check that the slope and y-intercept are correct.
Check that the equation is in the correct form.

Q: What are some resources for learning more about functions with absolute value?

A: Some resources for learning more about functions with absolute value include:

Online tutorials and videos.
Math textbooks and workbooks.
Online communities and forums.

Q: How do I practice graphing functions with absolute value?

A: To practice graphing functions with absolute value, you need to follow these steps:

Practice graphing different functions with absolute value.
Use online tools and software to graph functions with absolute value.
Work with a partner or tutor to practice graphing functions with absolute value.

Q: What are some tips for mastering functions with absolute value?

A: Some tips for mastering functions with absolute value include:

Practice graphing different functions with absolute value.
Use online tools and software to graph functions with absolute value.
Work with a partner or tutor to practice graphing functions with absolute value.

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

A: To apply functions with absolute value to real-world problems, you need to follow these steps:

Identify the problem and determine the values of $h$ and $k$.
Write the equation in the form $f(x) = |x - h| + k$.
Solve the equation to find the final answer.

Q: What are some common applications of functions with absolute value in science and engineering?

A: Some common applications of functions with absolute value in science and engineering include:

Modeling the distance between two points.
Modeling the temperature difference between two locations.
Modeling the cost of a product based on its weight.

Q: How do I use functions with absolute value in science and engineering?

A: To use functions with absolute value in science and engineering, you need to follow these steps:

Identify the problem and determine the values of $h$ and $k$.
Write the equation in the form $f(x) = |x - h| + k$.
Solve the equation to find the final answer.

Q: What are some common mistakes to avoid when using functions with absolute value in science and engineering?

A: Some common mistakes to avoid when using functions with absolute value in science and engineering include:

Not considering the two cases for graphing.
Not determining the slope and y-intercept correctly.
Not connecting the two linear segments at the vertex.

Q: How do I check my work when using functions with absolute value in science and engineering?

A: To check your work when using functions with absolute value in science and engineering, you need to follow these steps:

Verify that the two linear segments are connected at the vertex.
Check that the slope and y-intercept are correct.
Check that the equation is in the correct form.

Q: What are some resources for learning more about functions with absolute value in science and engineering?

A: Some resources for learning more about functions with absolute value in science and engineering include:

Online tutorials and videos.
Math textbooks and workbooks.
Online communities and forums.

Q: How do I practice using functions with absolute value in science and engineering?

A: To practice using functions with absolute value in science and engineering, you need to follow these steps:

Practice graphing different functions with absolute value.
Use online tools and software to graph functions with absolute value.
Work with a partner or tutor to practice graphing functions with absolute value.

Q: What are some tips for mastering functions with absolute value in science and engineering?

A: Some tips for mastering functions with absolute value in science and engineering include:

Practice graphing different functions with absolute value.
Use online tools and software to graph functions with absolute value.
Work with a partner or tutor to practice graphing functions with absolute value.