The graph of $h(x)=|x-10|+6$ is shown. On which interval is this graph increasing?
A. $(-\infty, 6)$
B. $(-\infty, 10)$
C. $(6, \infty)$
D. $(10, \infty)$

The function h ( x ) = ∣ x − 10∣ + 6 is an absolute value function with a vertex at x = 10 .
To the left of the vertex ( x < 10 ), the function is decreasing.
To the right of the vertex ( 10"> x > 10 ), the function is increasing.
The function is increasing on the interval ( 10 , ∞ ) ​ .

Explanation

Analyze the function We are given the function h ( x ) = ∣ x − 10∣ + 6 and asked to find the interval where the graph of this function is increasing. The absolute value function has a vertex where the expression inside the absolute value is equal to zero. In this case, the vertex occurs at x = 10 .

Determine the behavior to the left of the vertex To the left of the vertex (i.e., when x < 10 ), the function can be written as h ( x ) = − ( x − 10 ) + 6 = − x + 10 + 6 = − x + 16 . This is a linear function with a negative slope, so the function is decreasing on the interval ( − ∞ , 10 ) .

Determine the behavior to the right of the vertex To the right of the vertex (i.e., when 10"> x > 10 ), the function can be written as h ( x ) = ( x − 10 ) + 6 = x − 10 + 6 = x − 4 . This is a linear function with a positive slope, so the function is increasing on the interval ( 10 , ∞ ) .

Conclusion Therefore, the graph of h ( x ) is increasing on the interval ( 10 , ∞ ) .


Examples
Understanding where a function increases or decreases is crucial in many real-world applications. For example, if h ( x ) represents the profit of a company based on the number of units produced ( x ), knowing the interval where h ( x ) is increasing tells us the range of production levels where the company's profit is growing. Similarly, in physics, if h ( x ) represents the height of an object over time, the increasing interval indicates when the object is moving upwards.

Answered by GinnyAnswer | 2025-07-06

The graph of the function h ( x ) = ∣ x − 10∣ + 6 is increasing on the interval ( 10 , ∞ ) . Therefore, the correct answer is option D. ( 10 , ∞ ) .
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Answered by Anonymous | 2025-07-13