MAXIMUM AND MINIMUM VALUES
Definition Of Maximum And Minimum Values
Maximum Value: The maximum value of a quadratic function f(x) = ax2 + bx + c where a < 0,="" is="" the="" y-="" coordinate="" of="" the="">
Minimum Value: The minimum value of a quadratic function f(x) = ax2 + bx + c where a > 0, is the y- coordinate of the vertex.
Examples of Maximum and Minimum Values
The quadratic function f(x) = 9 - x2 has the maximum value of 9.
The quadratic function f(x) = 16 + x2 has the minimum value of 16.
Video Examples: Maximum and Minimum Values of Sine and Cosine Functions, Ex 1
Solved Example on Maximum and Minimum Values
Ques: Which of the following statements are correct for the function
f(x) = (x - 3)(x + 2) on the interval [- 5, 2]?
1. f(x) has Maximum Value at x = - 5
2. f(x) has Minimum Value at x = 2
3. f(x) has Maximum Value at x = - 3
4. f(x) has Minimum Value at x = 0.5
Choices:
A. 2 and 3 only
B. 1 and 3 only
C. 2 and 4 only
D. 1 and 4 only
Correct Answer: D
Solution:
Step 1: f(x) = (x - 3)(x + 2) [Write the function.]
Step 2: [Draw the graph of f(x) on the interval [- 5, 2].]
Step 3: f(x) has Maximum Value at x = - 5. [From the graph.]
Step 4: f(x) has Minimum Value at x = 0.5. [From the graph.]
Quick Summary
- If a < 0, the quadratic function has a maximum value at its vertex.
- If a > 0, the quadratic function has a minimum value at its vertex.
- The vertex's y-coordinate represents the maximum or minimum value.
🍎 Teacher Insights
Emphasize the connection between the graph of the quadratic function and its maximum/minimum value. Use real-world examples like projectile motion to illustrate the concept.🎓 Prerequisites
- Graphing quadratic functions
- Understanding the vertex of a parabola
- Solving quadratic equations
Check Your Knowledge
Q1: Which of the following functions has a maximum value?
Q2: The function f(x) = (x-3)(x+5) has a minimum value at which x-coordinate?
Frequently Asked Questions
Q: How do I find the maximum or minimum value of a quadratic function?
A: Find the vertex of the parabola. The y-coordinate of the vertex is the maximum or minimum value.
Q: What does the sign of 'a' tell me about the function?
A: If 'a' is positive, the parabola opens upwards and has a minimum value. If 'a' is negative, the parabola opens downwards and has a maximum value.