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Increasing Decreasing Functions Graphically

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Increasing Decreasing Functions Graphically
Practice Paper 1

Graph showing function with increasing and decreasing intervals

📊 Part 1: Interpreting Derivatives from Graphs

Success Criteria: I can identify where f'(x) > 0, f'(x) = 0, and f'(x) < 0 from a function's graph.

1. Look at the graph of f(x). Where is f'(x) > 0 (function is increasing)?

x < -2

-2 < x < 1

x > 1

-1 < x < 3

2. At which points is f'(x) = 0 (horizontal tangent)?

x = -2

x = 0

x = 1

x = 3

3. If a function has a local maximum at x = 2, what can you say about f'(2)?

f'(2) > 0

f'(2) = 0

f'(2) < 0

Cannot be determined

4. Where is the function decreasing most rapidly (steepest negative slope)?

✏️ Part 2: Analysing Function Behaviour

5. Complete the statements about the relationship between f(x) and f'(x):

When f'(x) > 0, the function f(x) is _______________.

When f'(x) < 0, the function f(x) is _______________.

When f'(x) = 0, the function f(x) has a _______________ tangent line.

6. Sketch a function that satisfies these conditions:

• f'(x) > 0 for x < -1

• f'(x) = 0 at x = -1

• f'(x) < 0 for x > -1

7. A ball is thrown upward. Its height function h(t) has h'(t) > 0 for 0 < t < 3 and h'(t) < 0 for t > 3. Explain what this means in context.

🎯 Part 3: Extension Challenge

8. Advanced: If f'(x) = x² - 4, find the intervals where f(x) is increasing and decreasing.
9. Differentiation Support: Match the graph descriptions to the derivative conditions:
A. Rising from left to right
B. Falling from left to right
C. Flat (horizontal) line
1. f'(x) = 0
2. f'(x) > 0
3. f'(x) < 0

Dyslexia-Friendly Tip: Remember the simple rule - positive derivative means going up, negative derivative means going down, zero derivative means flat!

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