Question

indicate which intervals are increasing, decreasing, or constant.
which intervals, if any, are increasing? select all that apply
a. (1) b. (3)
c. (6) d. (2)
e. (4) f. (5)
g. none of them
which intervals, if any, are decreasing? select all that apply
a. (5) b. (6)
c. (3) d. (2)
e. (4) f. (1)
g. none of them

Answer:

For the "Which intervals, if any, are decreasing?" question:

To determine decreasing intervals, we analyze the graph's slope:

• A function is decreasing when as \( x \) increases, \( y \) decreases.
• Interval (3): As \( x \) moves from left to right in interval (3), the \( y \)-values increase (so not decreasing).
• Interval (6): As \( x \) moves from left to right in interval (6), the \( y \)-values decrease (so this is a decreasing interval).
• Interval (2): As \( x \) moves from left to right in interval (2), the \( y \)-values decrease (wait, let's re-examine the graph). Wait, looking at the graph:
• Interval (2): From the root to the minimum, \( y \) decreases (so interval (2) is decreasing).
• Interval (6): From the peak to the right end, \( y \) decreases (so interval (6) is decreasing). Wait, maybe I misread earlier. Let's re-express:

Wait, the graph:

• Interval (1): From left end to the peak, \( y \) increases (so increasing, which matches the first question’s check on A).
• Interval (2): From peak to minimum, \( y \) decreases (so decreasing).
• Interval (3): From minimum to the next point, \( y \) increases (so increasing, which matches the first question’s check on B? Wait no, the first question’s options: A (1), B (3), F (5). Wait, maybe the intervals are labeled as (1), (2), (3), (4), (5), (6) on the x-axis. Let's re-express:

For decreasing intervals:

• Interval (2): As \( x \) increases, \( y \) decreases (so decreasing).
• Interval (6): As \( x \) increases, \( y \) decreases (so decreasing).
• Wait, but the options for the decreasing question: A (5), B (6), C (3), D (2), E (4), F (1), G (None).

So let's check each:

• A (5): Interval (5) – as \( x \) increases, \( y \) increases (so not decreasing).
• B (6): Interval (6) – as \( x \) increases, \( y \) decreases (so decreasing).
• C (3): Interval (3) – as \( x \) increases, \( y \) increases (so not decreasing).
• D (2): Interval (2) – as \( x \) increases, \( y \) decreases (so decreasing).
• E (4): Interval (4) – constant (horizontal line), so not decreasing.
• F (1): Interval (1) – increasing, so not decreasing.

Wait, but the original user’s screenshot shows B (6) is checked, but maybe D (2) is also decreasing. Wait, maybe the graph’s interval (2) is from the positive root to the minimum (where \( y \) goes from 0 down to the minimum), so as \( x \) increases, \( y \) decreases. So interval (2) is decreasing, and interval (6) is decreasing.

But let's confirm:

For the decreasing intervals:

• Interval (2): Decreasing (slope negative).
• Interval (6): Decreasing (slope negative).

But the options given:

Which intervals, if any, are decreasing? Select all that apply:

Options:
A. (5) – No (increasing)
B. (6) – Yes (decreasing)
C. (3) – No (increasing)
D. (2) – Yes (decreasing)
E. (4) – No (constant)
F. (1) – No (increasing)
G. None – No

So the correct answers for decreasing are B (6) and D (2). Wait, but in the user’s screenshot, B (6) is checked, but D (2) is not. Let's re-express the graph:

Looking at the left part:

• Interval (1): From \( x=-6 \) (approx) to \( x=-3 \) (peak) – \( y \) increases (so increasing, A is correct).
• Interval (2): From \( x=-3 \) (peak) to \( x=0 \) (root) to \( x=3 \) (minimum) – \( y \) decreases (so decreasing, D is correct).
• Interval (3): From \( x=3 \) (minimum) to \( x=6 \) (next peak) – \( y \) increases (so increasing, B is correct? Wait no, the first question’s options for increasing: A (1), B (3), F (5). So interval (3) is increasing, interval (5) is increasing (from \( x=6 \) to \( x=9 \), \( y \) increases? Wait, the graph has a horizontal segment (4) from \( x=6 \) to \( x=9 \)? No, the labels: (1), (2), (3), (4), (5), (6) on the x-axis. Let's assume:

• (1): Left of the first peak – increasing.
• (2): From first peak to minimum – decreasing.
• (3): From minimum to next peak – increasing.
• (4): Horizontal (constant).
• (5): From horizontal to next peak – increasing.
• (6): From next peak to right end – decreasing.

So for decreasing intervals:

• (2): decreasing (D)
• (6): decreasing (B)

So the correct answers for "Which intervals, if any, are decreasing?" are B. (6) and D. (2).

For the "Which intervals, if any, are increasing?" question (already partially answered):

From the graph:

• (1): increasing (A)
• (3): increasing (B)
• (5): increasing (F)

So the correct options are A. (1), B. (3), F. (5).

Final Answers:

For "Which intervals, if any, are increasing?":

A. (1), B. (3), F. (5)

For "Which intervals, if any, are decreasing?":

B. (6), D. (2)

(Note: The user’s screenshot shows some pre-checked boxes, but the correct analysis gives these as the right selections.)