Question

1.3 increasing and decreasing intervals
approximate the intervals where each function is increasing and decreasing.

1. color the increasing intervals in: graph
2. color the decreasing intervals in: graph
3. graph
4. graph
5. graph
6. graph

Explanation:

To solve problems about identifying increasing and decreasing intervals of a function from its graph, we use the definition: a function is increasing on an interval if, as \( x \) increases (moves to the right), \( f(x) \) also increases (moves up). A function is decreasing on an interval if, as \( x \) increases, \( f(x) \) decreases (moves down).

Problem 1: Analyze the first graph (1)

Step 1: Identify Increasing Intervals

• Observe the graph: Find where the curve rises as \( x \) increases.
• From \( x = -2 \) to \( x = -1 \): The graph rises (increasing).
• From \( x = 0 \) to \( x = 1 \): The graph rises (increasing).

Step 2: Identify Decreasing Intervals

• Find where the curve falls as \( x \) increases.
• From \( x = -1 \) to \( x = 0 \): The graph falls (decreasing).

Problem 2: Analyze the second graph (2)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -3 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = 0 \) to \( x = 3 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 3: Analyze the third graph (3)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = 1 \) to \( x = 4 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -1 \) to \( x = 1 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 4: Analyze the fourth graph (4)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -2 \) to \( x = 2 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = 2 \) to \( x = 5 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 5: Analyze the fifth graph (5)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -2 \) to \( x = -1 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -3 \) to \( x = -2 \) and from \( x = -1 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 6: Analyze the sixth graph (6)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -4 \) to \( x = -3 \) and from \( x = -1 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -3 \) to \( x = -1 \) (as \( x \) increases, \( f(x) \) decreases).

Final Answers (Approximate Intervals)

1. Increasing: \( (-2, -1) \cup (0, 1) \); Decreasing: \( (-1, 0) \)
2. Increasing: \( (-3, 0) \); Decreasing: \( (0, 3) \)
3. Increasing: \( (1, 4) \); Decreasing: \( (-1, 1) \)
4. Increasing: \( (-2, 2) \); Decreasing: \( (2, 5) \)
5. Increasing: \( (-2, -1) \); Decreasing: \( (-3, -2) \cup (-1, 0) \)
6. Increasing: \( (-4, -3) \cup (-1, 0) \); Decreasing: \( (-3, -1) \)

(Note: Intervals are approximate based on the grid; adjust for precise graph details.)

Answer:

To solve problems about identifying increasing and decreasing intervals of a function from its graph, we use the definition: a function is increasing on an interval if, as \( x \) increases (moves to the right), \( f(x) \) also increases (moves up). A function is decreasing on an interval if, as \( x \) increases, \( f(x) \) decreases (moves down).

Problem 1: Analyze the first graph (1)

Step 1: Identify Increasing Intervals

• Observe the graph: Find where the curve rises as \( x \) increases.
• From \( x = -2 \) to \( x = -1 \): The graph rises (increasing).
• From \( x = 0 \) to \( x = 1 \): The graph rises (increasing).

Step 2: Identify Decreasing Intervals

• Find where the curve falls as \( x \) increases.
• From \( x = -1 \) to \( x = 0 \): The graph falls (decreasing).

Problem 2: Analyze the second graph (2)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -3 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = 0 \) to \( x = 3 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 3: Analyze the third graph (3)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = 1 \) to \( x = 4 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -1 \) to \( x = 1 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 4: Analyze the fourth graph (4)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -2 \) to \( x = 2 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = 2 \) to \( x = 5 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 5: Analyze the fifth graph (5)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -2 \) to \( x = -1 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -3 \) to \( x = -2 \) and from \( x = -1 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) decreases).

Problem 6: Analyze the sixth graph (6)

Step 1: Identify Increasing Intervals

• The graph rises from \( x = -4 \) to \( x = -3 \) and from \( x = -1 \) to \( x = 0 \) (as \( x \) increases, \( f(x) \) increases).

Step 2: Identify Decreasing Intervals

• The graph falls from \( x = -3 \) to \( x = -1 \) (as \( x \) increases, \( f(x) \) decreases).

Final Answers (Approximate Intervals)

1. Increasing: \( (-2, -1) \cup (0, 1) \); Decreasing: \( (-1, 0) \)
2. Increasing: \( (-3, 0) \); Decreasing: \( (0, 3) \)
3. Increasing: \( (1, 4) \); Decreasing: \( (-1, 1) \)
4. Increasing: \( (-2, 2) \); Decreasing: \( (2, 5) \)
5. Increasing: \( (-2, -1) \); Decreasing: \( (-3, -2) \cup (-1, 0) \)
6. Increasing: \( (-4, -3) \cup (-1, 0) \); Decreasing: \( (-3, -1) \)

(Note: Intervals are approximate based on the grid; adjust for precise graph details.)