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What is the Maximum/Minimum Value of a Quadratic Function?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The maximum or minimum value of a quadratic function is the highest or lowest point its graph reaches. For a U-shaped curve (parabola), this point is called the vertex, and it represents either the peak (maximum) or the bottom (minimum) value the function can have.
Simple Example
Quick Example
Imagine a cricket ball hit high in the air. Its path forms a curve. The highest point the ball reaches before coming down is its maximum height. This maximum height is like the maximum value of a quadratic function describing the ball's path.
Worked Example
Step-by-Step
Let's find the minimum value of the quadratic function y = x^2 - 4x + 5.
Step 1: Identify the coefficients a, b, and c. Here, a=1, b=-4, c=5.
---Step 2: Since 'a' is positive (a=1 > 0), the parabola opens upwards, meaning it will have a minimum value.
---Step 3: The x-coordinate of the vertex (where the minimum occurs) is given by the formula x = -b / (2a).
---Step 4: Substitute the values: x = -(-4) / (2 * 1) = 4 / 2 = 2.
---Step 5: Now, substitute this x-value (x=2) back into the original function to find the minimum y-value.
---Step 6: y = (2)^2 - 4(2) + 5 = 4 - 8 + 5 = 1.
---The minimum value of the quadratic function y = x^2 - 4x + 5 is 1.
Why It Matters
Understanding maximum and minimum values helps scientists predict the best launch angle for rockets or engineers design bridges. Data scientists use this to find optimal solutions in algorithms, and economists predict peak profits or lowest costs for businesses. It's crucial for careers in AI, engineering, and finance.
Common Mistakes
MISTAKE: Confusing maximum with minimum. If 'a' is positive, students might incorrectly look for a maximum. | CORRECTION: Remember, if 'a' (the coefficient of x^2) is positive, the parabola opens upwards and has a MINIMUM. If 'a' is negative, it opens downwards and has a MAXIMUM.
MISTAKE: Forgetting to substitute the x-value back into the original function. Students often stop after finding the x-coordinate of the vertex. | CORRECTION: The question asks for the MAXIMUM/MINIMUM VALUE, which is the y-coordinate. Always substitute the x-coordinate back into the function to find 'y'.
MISTAKE: Making calculation errors with negative signs in the formula x = -b / (2a). | CORRECTION: Be extra careful with double negatives, like -(-4) becoming +4. Always write out the substitution clearly.
Practice Questions
Try It Yourself
QUESTION: Find the minimum value of the function y = x^2 - 6x + 10. | ANSWER: 1
QUESTION: Find the maximum value of the function y = -x^2 - 2x + 3. | ANSWER: 4
QUESTION: A small shop's daily profit (in Rupees) is given by the function P(x) = -2x^2 + 20x - 30, where 'x' is the number of items sold. What is the maximum profit the shop can make? | ANSWER: 20 Rupees
MCQ
Quick Quiz
For a quadratic function y = ax^2 + bx + c, if a < 0, what kind of value does it have?
A minimum value
A maximum value
Both a maximum and a minimum value
Neither a maximum nor a minimum value
The Correct Answer Is:
B
If 'a' is negative (a < 0), the parabola opens downwards, like an inverted U, meaning its highest point is the maximum value. It goes infinitely downwards.
Real World Connection
In the Real World
Farmers in India use this concept to maximize crop yield or minimize costs. For instance, if the yield of a crop depends quadratically on the amount of fertilizer used, they can use this math to find the 'sweet spot' for fertilizer application to get the highest possible harvest. This helps them earn more from their hard work.
Key Vocabulary
Key Terms
QUADRATIC FUNCTION: A function of the form y = ax^2 + bx + c | VERTEX: The highest or lowest point on the graph of a quadratic function | PARABOLA: The U-shaped curve that is the graph of a quadratic function | COEFFICIENT: A number multiplied by a variable in an algebraic expression
What's Next
What to Learn Next
Great job understanding the vertex of a quadratic function! Next, you can explore 'Graphing Quadratic Functions'. Knowing the maximum/minimum point will make sketching these graphs much easier and more accurate. Keep learning!
