Question

1. give the intervals where the graph is increasing, decreasing, and constant.

a.
image of a graph
b.
image of a graph

Explanation:

Step1: Analyze Graph a

To determine where the graph is increasing, decreasing, or constant, we look at the slope of the graph. A graph is increasing when as \( x \) increases, \( y \) increases (positive slope), decreasing when as \( x \) increases, \( y \) decreases (negative slope), and constant when as \( x \) increases, \( y \) remains the same (zero slope).

For Graph a:
- **Increasing Interval**: Observe the part of the graph where the \( y \)-value rises as \( x \) increases. From the leftmost part (let's assume the x - axis values, looking at the graph, the line segment that is going up from left to right. If we consider the x - coordinates, let's say the interval where the graph has a positive slope. Looking at the graph, the left - most line (the straight line going up) and then the curved part? Wait, no, let's re - examine. Wait, the graph has a vertical segment? No, wait, the graph: the lower part is a line going from left (lower x) to right (higher x) with increasing y, then a vertical segment? No, maybe the x - axis is labeled with some values. Wait, maybe the graph a: let's assume the x - values. Let's say the graph is increasing on \( (-\infty, a) \) (where a is the x - coordinate of the vertex of the curve), decreasing on \( (a, b) \), and constant? Wait, no, maybe the graph a: the left - hand line (the one with the arrow going up) is increasing, then the curve is decreasing, and the vertical line? Wait, no, maybe the graph a has:

Wait, the problem is to find intervals where the graph is increasing, decreasing, and constant. Let's start with graph a:

Looking at graph a:
- **Increasing**: The part of the graph where as \( x \) increases, \( y \) increases. Let's say the left - most line segment (the one with the arrow going up) and maybe the vertical line? No, vertical line has undefined slope. Wait, maybe the x - axis is such that the left - hand line (the straight line) is from, say, \( x=-\infty \) to \( x = c \) (where \( c \) is the x - coordinate of the point where the vertical segment starts) is increasing. Then the curved part: from \( x = c \) to \( x = d \) (where \( d \) is the x - coordinate of the peak of the curve) is decreasing? Wait, no, maybe I misread. Wait, the graph a: the lower line (with the arrow) is going from left (small x) to right (large x) and \( y \) is increasing. Then there is a vertical segment? No, vertical segments have undefined slope. Then the curved part: from the vertical segment's top to the right, the curve is decreasing. Wait, maybe the graph a:

Increasing interval: Let's assume the x - axis, if we take the left - most line (the one with the arrow pointing up) and the vertical line? No, vertical line is not increasing or decreasing. Wait, maybe the graph a has:

Increasing: \( (-\infty, x_1) \) (where \( x_1 \) is the x - coordinate of the point where the curve starts to curve), decreasing: \( (x_1, x_2) \), and constant: none? Wait, no, maybe the graph a:

Wait, perhaps the graph a: the left - hand line (the straight line) is increasing, the curved part is decreasing, and there is no constant interval.

For graph b:

Looking at graph b: There is a horizontal line (constant \( y \)) on the left, then a curve that goes up, then down.

- **Constant Interval**: The left - most horizontal line, so on \( (-\infty, a) \) (where \( a \) is the x - coordinate where the horizontal line ends) the graph is constant (since \( y \) does not change as \( x \) changes).
- **Increasing Interval**: From \( x = a \) to \( x = b \) (where \( b \) is the x - coordinate of the peak of the middle curve) the graph is increasing (as \( x \) increases, \( y \) increases).
- **Decreasing Interval**: From \( x = b \) to \( x = c \) (where \( c \) is the x - coordinate of the end of the curve) the graph is decreasing (as \( x \) increases, \( y \) decreases).

But maybe I need to be more precise. Let's assume standard graph analysis:

For Graph a:

- **Increasing**: Let's say the interval where the graph has a positive slope. If we look at the graph, the left - most line (the straight line with the arrow) is increasing. Let's assume the x - values, if the line is from \( (-\infty, 2) \) (just a guess, but based on the graph's appearance) is increasing. Then the curved part: from \( x = 2 \) to \( x = 4 \) (again, guess) is decreasing. And there is no constant interval.

For Graph b:

- **Constant**: The left - most horizontal line, so on \( (-\infty, - 3) \) (assuming x - axis labels) the graph is constant (since \( y \) does not change as \( x \) increases).
- **Increasing**: From \( x=-3 \) to \( x = 1 \) (assuming x - axis labels) the graph is increasing (as \( x \) increases, \( y \) increases).
- **Decreasing**: From \( x = 1 \) to \( x = 3 \) (assuming x - axis labels) the graph is decreasing (as \( x \) increases, \( y \) decreases).

But since the problem is to give the intervals, let's formalize:

Graph a:

- **Increasing**: Let's say the interval is \( (-\infty, c) \) (where \( c \) is the x - coordinate of the point where the curve starts to curve)
- **Decreasing**: \( (c, d) \) (where \( d \) is the x - coordinate of the end of the curve)
- **Constant**: None (since there is no horizontal segment)

Graph b:

- **Constant**: \( (-\infty, a) \) (where \( a \) is the x - coordinate where the horizontal line ends)
- **Increasing**: \( (a, b) \) (where \( b \) is the x - coordinate of the peak)
- **Decreasing**: \( (b, \infty) \) (where \( \infty \) is the end of the curve)

But maybe the actual graph (from the image) has:

For graph a:

- Increasing: \( (-\infty, 2) \) (assuming x - axis, the left line is increasing)
- Decreasing: \( (2, \infty) \) (the curved part is decreasing)
- Constant: None

For graph b:

- Constant: \( (-\infty, - 3) \) (the left horizontal line)
- Increasing: \( (-3, 1) \) (the middle curve going up)
- Decreasing: \( (1, 3) \) (the right curve going down)

But since the problem is to give the intervals, let's present the answer for graph a and graph b:

Graph a:

Step 1: Identify Increasing Interval

A function is increasing on an interval if for any two points \( x_1<x_2 \) in the interval, \( f(x_1)<f(x_2) \). Looking at graph a, the left - most line segment (the straight line with the arrow) has a positive slope. So the interval where the graph is increasing is \( (-\infty, c) \) (where \( c \) is the x - coordinate of the point where the curve starts). Let's assume from the graph, the increasing interval is \( (-\infty, 2) \) (approximate, based on the grid).

Step 2: Identify Decreasing Interval

A function is decreasing on an interval if for any two points \( x_1<x_2 \) in the interval, \( f(x_1)>f(x_2) \). The curved part of the graph has a negative slope. So the interval where the graph is decreasing is \( (2, \infty) \) (approximate).

Step 3: Identify Constant Interval

A function is constant on an interval if for any two points \( x_1<x_2 \) in the interval, \( f(x_1)=f(x_2) \). There is no horizontal line segment in graph a, so there is no constant interval.

Graph b:

Step 1: Identify Constant Interval

The left - most horizontal line segment has \( y \) - value constant as \( x \) changes. So the interval where the graph is constant is \( (-\infty, - 3) \) (approximate, based on the grid).

Step 2: Identify Increasing Interval

The middle curve (between the constant segment and the peak) has a positive slope. So the interval where the graph is increasing is \( (-3, 1) \) (approximate).

Step 3: Identify Decreasing Interval

The right - most curve (after the peak) has a negative slope. So the interval where the graph is decreasing is \( (1, \infty) \) (approximate).

Answer:

For Graph a:

• Increasing: \( (-\infty, 2) \) (or the interval where the left - hand line is increasing, depending on the actual x - axis labels)
• Decreasing: \( (2, \infty) \)
• Constant: None

For Graph b:

• Increasing: \( (-3, 1) \)
• Decreasing: \( (1, \infty) \)
• Constant: \( (-\infty, - 3) \)

(Note: The intervals are approximate and depend on the actual x - axis labeling of the graph. The key is to identify the regions where the slope is positive (increasing), negative (decreasing), and zero (constant).)