Question

match each graph on the left with an interval where the function is constant.

Answer:

To solve this problem, we analyze each graph to identify the interval where the function is constant (horizontal line segment, meaning the \( y \)-value doesn't change as \( x \) changes):

First Graph (Top - Left):

This graph has a "V" - shaped part and a curved part. Let's check the intervals. The left part is a line decreasing, then a minimum, then a curve increasing. There's no horizontal segment here? Wait, maybe I mis - look. Wait, no, let's re - examine. Wait, the first graph: the left side is a line from a point, decreasing to a vertex, then the right side is a curve increasing. So no constant interval? Wait, no, maybe the intervals on the right are given. Wait, maybe the second graph (middle left) has a horizontal segment.

Second Graph (Middle - Left):

This graph has a curve on the left (a "U" - shaped part), then a vertical jump? No, wait, after the "U" - shaped part, there's a horizontal line segment (constant \( y \) - value as \( x \) increases). So this graph's constant interval is where the line is horizontal.

Third Graph (Bottom - Left):

This graph has a "W" - like shape with line segments. Let's check for horizontal parts. The middle part: from a peak, does it stay constant? No, it's a line going up then down? Wait, no, maybe the third graph has a horizontal segment? Wait, no, let's think about the definition: a function is constant on an interval if for all \( x_1, x_2 \) in the interval, \( f(x_1)=f(x_2) \), which means the graph is a horizontal line on that interval.

Let's match:

• The middle - left graph (with the "U" and then a horizontal line) should be matched to the interval where the function is constant (the horizontal segment).
• The bottom - left graph: let's see, if there's a horizontal part? Wait, maybe the bottom - left graph has a horizontal segment? Wait, no, the bottom - left graph is made of line segments with slopes (not horizontal). Wait, maybe the first graph: no, the first graph's right part is a curve. Wait, perhaps the intervals on the right (the dashed boxes) have different intervals. Since the problem is about matching graphs to intervals where the function is constant, we look for the graph with a horizontal line segment.

The middle - left graph has a horizontal line segment (constant \( y \) as \( x \) increases), so we match it to the interval corresponding to that horizontal part. The bottom - left graph: maybe no, wait, the third graph (bottom - left) – does it have a horizontal segment? No, it's a polygonal graph with changing slopes. The first graph: no. So the middle - left graph (with the "U" and then horizontal) is matched to the interval where \( x \) is in a range with constant \( y \).

But since the problem is about matching, let's assume the intervals on the right are, for example, \( [a,b] \) where the function is constant. The middle - left graph has a horizontal segment, so we match it to the interval with the horizontal line.

(Note: Since the right - hand intervals are not clearly visible (the image has a light effect), but based on the standard problem of "function constant on an interval" (horizontal line segment in the graph), we identify the graph with a horizontal line segment and match it to the corresponding interval. The middle - left graph has a horizontal line segment, so it should be matched to the interval where the function is constant.)

If we assume the intervals on the right are, say:

1. An interval for the middle - left graph (horizontal segment)
2. An interval for another graph
3. An interval for the third graph

But since the exact intervals are not fully visible, the key is to find the graph with a horizontal line (constant function on that interval) and match it. The middle - left graph (with the "U" and then a horizontal line) is the one with a constant interval.