Question

the graph of $h(x)=|x - 10|+6$ is shown. on which interval is this graph increasing? $\bigcirc\\ (-infty, 6)$ $\bigcirc\\ (-infty, 10)$ $\bigcirc\\ (6, infty)$ $\bigcirc\\ (10, infty)$

Explanation:

Step1: Recall absolute value function shape

The function \( h(x) = |x - 10| + 6 \) is an absolute value function. The general form of an absolute value function is \( f(x)=|x - h|+k \), where \((h,k)\) is the vertex. For \( h(x)=|x - 10|+6 \), the vertex is \((10,6)\).

Step2: Analyze the slope of the graph

The graph of an absolute value function \( y = |x - a|+b \) has a "V" shape. To the left of the vertex (\( x < a \)), the slope is \(- 1\) (decreasing), and to the right of the vertex (\( x > a \)), the slope is \(1\) (increasing). Here, \( a = 10 \), so the function is increasing when \( x>10 \), i.e., on the interval \((10,\infty)\).

Answer:

\( (10, \infty) \) (the option with this interval, e.g., if the options are as given, the correct one is the last option: \( (10, \infty) \))