The Graph Of $h(x)=|x-10|+6$ Is Shown. On Which Interval Is This Graph Increasing?A. ( − ∞ , 6 (-\infty, 6 ( − ∞ , 6 ] B. ( − ∞ , 10 (-\infty, 10 ( − ∞ , 10 ] C. ( 6 , ∞ (6, \infty ( 6 , ∞ ] D. ( 10 , ∞ (10, \infty ( 10 , ∞ ]

Introduction

When analyzing the graph of a function, it's essential to understand the behavior of the function, including where it is increasing or decreasing. In this article, we will focus on the graph of the function $h(x)=|x-10|+6$ and determine the interval on which this graph is increasing.

Understanding the Function

The given function is $h(x)=|x-10|+6$. This function involves the absolute value of the difference between $x$ and $10$, added to $6$. To understand the behavior of this function, we need to consider the different cases based on the value of $x$.

Case 1: $x < 10$

When $x < 10$, the expression $|x-10|$ simplifies to $-(x-10)$, which is equivalent to $-x+10$. Therefore, for $x < 10$, the function becomes $h(x)=-x+10+6=-x+16$.

Case 2: $x \geq 10$

When $x \geq 10$, the expression $|x-10|$ simplifies to $x-10$. Therefore, for $x \geq 10$, the function becomes $h(x)=x-10+6=x-4$.

Analyzing the Function

To determine the interval on which the graph of $h(x)$ is increasing, we need to find the critical points of the function. The critical points occur when the derivative of the function is equal to zero or undefined.

Derivative of the Function

To find the derivative of the function, we will use the power rule and the sum rule.

For $x < 10$, the derivative of $h(x)=-x+16$ is $h'(x)=-1$.

For $x \geq 10$, the derivative of $h(x)=x-4$ is $h'(x)=1$.

Critical Points

The critical points occur when the derivative of the function is equal to zero or undefined. In this case, the derivative is a constant, so there are no critical points.

Increasing Interval

Since the derivative of the function is a constant, the function is either increasing or decreasing on its entire domain. To determine the interval on which the graph of $h(x)$ is increasing, we need to examine the behavior of the function.

Behavior of the Function

For $x < 10$, the function $h(x)=-x+16$ is decreasing, since the derivative $h'(x)=-1$ is negative.

For $x \geq 10$, the function $h(x)=x-4$ is increasing, since the derivative $h'(x)=1$ is positive.

Conclusion

Based on the analysis of the function, we can conclude that the graph of $h(x)$ is increasing on the interval $(10, \infty)$.

Final Answer

The final answer is $\boxed{(10, \infty)}$.

Discussion

The graph of the function $h(x)=|x-10|+6$ is increasing on the interval $(10, \infty)$. This is because the derivative of the function is positive on this interval, indicating that the function is increasing.

Related Topics

• Graphing functions
• Analyzing functions
• Critical points
• Increasing and decreasing intervals

References

• [1] Calculus: Early Transcendentals, James Stewart
• [2] Calculus, Michael Spivak

Note: The references provided are for general calculus textbooks and are not specific to the topic of this article.

Introduction

In our previous article, we analyzed the graph of the function $h(x)=|x-10|+6$ and determined that the graph is increasing on the interval $(10, \infty)$. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q1: What is the definition of an increasing function?

A1: An increasing function is a function whose value increases as the input increases. In other words, if $x_1 < x_2$, then $f(x_1) < f(x_2)$.

Q2: How do you determine if a function is increasing or decreasing?

A2: To determine if a function is increasing or decreasing, you need to examine the behavior of the function. You can do this by finding the derivative of the function and analyzing its sign.

Q3: What is the significance of the critical points in a function?

A3: Critical points are the points where the derivative of the function is equal to zero or undefined. These points are significant because they can indicate where the function changes from increasing to decreasing or vice versa.

Q4: How do you find the derivative of a function?

A4: To find the derivative of a function, you can use the power rule, the sum rule, and the product rule. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. The sum rule states that if $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$. The product rule states that if $f(x) = g(x)h(x)$, then $f'(x) = g'(x)h(x) + g(x)h'(x)$.

Q5: What is the relationship between the derivative of a function and its graph?

A5: The derivative of a function represents the slope of the tangent line to the graph of the function at a given point. If the derivative is positive, the graph is increasing. If the derivative is negative, the graph is decreasing.

Q6: How do you determine the interval on which a function is increasing or decreasing?

A6: To determine the interval on which a function is increasing or decreasing, you need to examine the behavior of the function. You can do this by finding the derivative of the function and analyzing its sign.

Q7: What is the significance of the absolute value function in this problem?

A7: The absolute value function is used to represent the distance between two points on the number line. In this problem, the absolute value function is used to represent the distance between $x$ and $10$.

Q8: How do you simplify the absolute value function?

A8: To simplify the absolute value function, you can use the following rules:

• If $x \geq 0$, then $|x| = x$.
• If $x < 0$, then $|x| = -x$.

Q9: What is the relationship between the absolute value function and the graph of the function?

A9: The absolute value function is used to represent the graph of the function. The graph of the function is increasing on the interval $(10, \infty)$ because the derivative of the function is positive on this interval.

Q10: How do you determine the interval on which the graph of the function is increasing?

A10: To determine the interval on which the graph of the function is increasing, you need to examine the behavior of the function. You can do this by finding the derivative of the function and analyzing its sign.

Conclusion

In this article, we answered some frequently asked questions related to the graph of the function $h(x)=|x-10|+6$. We discussed the definition of an increasing function, how to determine if a function is increasing or decreasing, and the significance of critical points. We also discussed how to find the derivative of a function and the relationship between the derivative of a function and its graph.

Final Answer

The final answer is $\boxed{(10, \infty)}$.

Discussion

The graph of the function $h(x)=|x-10|+6$ is increasing on the interval $(10, \infty)$. This is because the derivative of the function is positive on this interval, indicating that the function is increasing.

Related Topics

• Graphing functions
• Analyzing functions
• Critical points
• Increasing and decreasing intervals
• Absolute value functions
• Derivatives

References

• [1] Calculus: Early Transcendentals, James Stewart
• [2] Calculus, Michael Spivak

Note: The references provided are for general calculus textbooks and are not specific to the topic of this article.