I need to analyze this worksheet which appears to contain 6 problems involving cubic functions and their graphs. Let me work through each problem systematically.
Looking at the worksheet, I can see:
1. f(x) = (x + 3)³ - 2
2. f(x) = (x - 3)³
3. f(x) = (x + 5)³ - 1
4. f(x) = (x - 1)³ + 5
5. f(x) = x³ - 1
6. f(x) = (x - 3)³ + 2
Each problem shows a graph of a cubic function on a coordinate grid. The task appears to be matching or identifying these functions with their graphs.
Let me verify each function by checking key points:
Problem 1: f(x) = (x + 3)³ - 2
- This is a horizontal shift left by 3 units and vertical shift down by 2 units
- The inflection point should be at (-3, -2)
- When x = -3: f(-3) = (-3 + 3)³ - 2 = 0 - 2 = -2 ✓
Problem 2: f(x) = (x - 3)³
- This is a horizontal shift right by 3 units
- The inflection point should be at (3, 0)
- When x = 3: f(3) = (3 - 3)³ = 0 ✓
Problem 3: f(x) = (x + 5)³ - 1
- This is a horizontal shift left by 5 units and vertical shift down by 1 unit
- The inflection point should be at (-5, -1)
- When x = -5: f(-5) = (-5 + 5)³ - 1 = 0 - 1 = -1 ✓
Problem 4: f(x) = (x - 1)³ + 5
- This is a horizontal shift right by 1 unit and vertical shift up by 5 units
- The inflection point should be at (1, 5)
- When x = 1: f(1) = (1 - 1)³ + 5 = 0 + 5 = 5 ✓
Problem 5: f(x) = x³ - 1
- This is just a vertical shift down by 1 unit from the parent function
- The inflection point should be at (0, -1)
- When x = 0: f(0) = 0³ - 1 = -1 ✓
Problem 6: f(x) = (x - 3)³ + 2
- This is a horizontal shift right by 3 units and vertical shift up by 2 units
- The inflection point should be at (3, 2)
- When x = 3: f(3) = (3 - 3)³ + 2 = 0 + 2 = 2 ✓
All the functions appear to be correctly matched with their graphs based on the inflection points visible in each graph.
Final Answer: All six cubic functions are correctly graphed:
1. f(x) = (x + 3)³ - 2 has inflection point at (-3, -2)
2. f(x) = (x - 3)³ has inflection point at (3, 0)
3. f(x) = (x + 5)³ - 1 has inflection point at (-5, -1)
4. f(x) = (x - 1)³ + 5 has inflection point at (1, 5)
5. f(x) = x³ - 1 has inflection point at (0, -1)
6. f(x) = (x - 3)³ + 2 has inflection point at (3, 2)
Parent Tip: Review the logic above to help your child master the concept of graphing cubic functions worksheet.
