Question

properties of functions
find the intervals where the function is increasing, decreasing, and constant, if any.

Explanation:

Step1: Recall function behavior rules

A function is increasing when the y - values increase as x - values increase, decreasing when y - values decrease as x - values increase, and constant when y - values do not change as x - values change.

Step2: Analyze graph 1

For the first parabola - shaped graph:

• It is decreasing on the interval $(-\infty, a)$ (where $a$ is the x - coordinate of the vertex).
• It is increasing on the interval $(a,\infty)$.
• There are no constant intervals.

Step3: Analyze graph 2

For the second graph (a curve starting from the left and going up):

• It is increasing on its entire domain, say $(-\infty,\infty)$.
• There are no decreasing or constant intervals.

Step4: Analyze graph 3

For the third graph (a straight - line with negative slope):

• It is decreasing on its entire domain, say $(-\infty,\infty)$.
• There are no increasing or constant intervals.

Step5: Analyze graph 4

For the fourth graph (a line segment going up and then horizontal):

• It is increasing on the first part of the domain, say $(-\infty, b)$ (where $b$ is the x - coordinate of the point where it becomes horizontal).
• It is constant on the interval $(b,\infty)$.
• There are no decreasing intervals.

Step6: Analyze graph 5

For the fifth graph (a zig - zag graph):

• It is increasing on intervals like $(c_1, c_2)$ and $(c_3, c_4)$ (where $c_i$ are x - coordinates of the turning points).
• It is decreasing on intervals like $(c_2, c_3)$ and $(c_4,\infty)$.
• There are no constant intervals.

Step7: Analyze graph 6

For the sixth graph (a piece - wise linear graph with increasing and then constant parts):

• It is increasing on the first two non - horizontal parts of the domain, say $(-\infty, d_1)$ and $(d_1, d_2)$ (where $d_i$ are x - coordinates of the turning points).
• It is constant on the interval $(d_2,\infty)$.
• There are no decreasing intervals.

Since we don't have exact coordinates from the graph, we'll just give general forms of intervals.

1.

• Increasing: $(a,\infty)$
• Decreasing: $(-\infty, a)$
• Constant: None

2.

• Increasing: $(-\infty,\infty)$
• Decreasing: None
• Constant: None

3.

• Increasing: None
• Decreasing: $(-\infty,\infty)$
• Constant: None

4.

• Increasing: $(-\infty, b)$
• Decreasing: None
• Constant: $(b,\infty)$

5.

• Increasing: Intervals between local minima and local maxima
• Decreasing: Intervals between local maxima and local minima
• Constant: None

6.

• Increasing: $(-\infty, d_1)\cup(d_1, d_2)$
• Decreasing: None
• Constant: $(d_2,\infty)$

(Here $a$, $b$, $d_1$, $d_2$ are x - coordinates of relevant turning points on the graphs which can be estimated from the grid on the graphs)

Answer:

1. Increasing: $(a,\infty)$; Decreasing: $(-\infty, a)$; Constant: None
2. Increasing: $(-\infty,\infty)$; Decreasing: None; Constant: None
3. Increasing: None; Decreasing: $(-\infty,\infty)$; Constant: None
4. Increasing: $(-\infty, b)$; Decreasing: None; Constant: $(b,\infty)$
5. Increasing: Intervals between local minima and local maxima; Decreasing: Intervals between local maxima and local minima; Constant: None
6. Increasing: $(-\infty, d_1)\cup(d_1, d_2)$; Decreasing: None; Constant: $(d_2,\infty)$