What is the Minimum Point of a Parabola?

S3-SA5-0279

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Minimum Point of a Parabola is the lowest point on its graph. It's also called the 'vertex' and represents the smallest possible value the parabola can reach.

Simple Example

Quick Example

Imagine a cricket ball hit high in the air. Its path forms a parabola. The highest point the ball reaches before coming down is like the 'maximum point' of a parabola that opens downwards. If the parabola opens upwards, its lowest point, like the bottom of a 'U' shape, is its minimum point.

Worked Example

Step-by-Step

Let's find the minimum point of the parabola given by the equation y = x^2 - 4x + 3.
---Step 1: Identify the coefficients. For y = ax^2 + bx + c, we have a=1, b=-4, c=3.
---Step 2: The x-coordinate of the minimum point (vertex) is given by the formula x = -b / (2a). Substitute the values: x = -(-4) / (2 * 1).
---Step 3: Calculate the x-coordinate: x = 4 / 2 = 2.
---Step 4: Substitute this x-value back into the original equation to find the y-coordinate. y = (2)^2 - 4(2) + 3.
---Step 5: Calculate the y-coordinate: y = 4 - 8 + 3 = -1.
---Step 6: The minimum point (vertex) is (x, y) = (2, -1).
Answer: The minimum point of the parabola y = x^2 - 4x + 3 is (2, -1).

Why It Matters

Understanding the minimum point is crucial in fields like AI/ML and Data Science to find optimal solutions or the lowest error rates. Engineers use it to design structures for maximum stability, and economists might use it to find the lowest production cost for a product, helping businesses save money.

Common Mistakes

MISTAKE: Confusing minimum point with maximum point | CORRECTION: A parabola opening upwards (like a 'U' shape, when 'a' is positive) has a minimum point. A parabola opening downwards (like an 'n' shape, when 'a' is negative) has a maximum point.

MISTAKE: Forgetting to find the y-coordinate after finding x | CORRECTION: Always substitute the calculated x-coordinate back into the original parabola equation to find the corresponding y-coordinate. The minimum point is always a pair (x, y).

MISTAKE: Incorrectly applying the formula x = -b / (2a) due to sign errors | CORRECTION: Pay close attention to the signs of 'b' when substituting into the formula. Remember -(-b) becomes +b.

Practice Questions

Try It Yourself

QUESTION: What is the x-coordinate of the minimum point for the parabola y = x^2 + 6x + 5? | ANSWER: x = -3

QUESTION: Find the minimum point of the parabola y = 2x^2 - 8x + 10. | ANSWER: (2, 2)

QUESTION: A company's daily profit (in thousands of rupees) from selling 'x' units of a new mobile charger is given by P(x) = x^2 - 10x + 30. What is the minimum profit the company can make, and how many units should they sell to achieve it? | ANSWER: Minimum profit is 5 thousand rupees when 5 units are sold.

MCQ

Quick Quiz

For a parabola given by y = 3x^2 - 12x + 15, which of the following statements is true about its minimum point?

The x-coordinate is -2

The parabola opens downwards

The minimum y-value is 3

The minimum point is (2, 15)

The Correct Answer Is:

For y = 3x^2 - 12x + 15, a=3 (positive), so it opens upwards and has a minimum. x = -(-12)/(2*3) = 12/6 = 2. y = 3(2)^2 - 12(2) + 15 = 12 - 24 + 15 = 3. So the minimum point is (2, 3), and the minimum y-value is 3.

Real World Connection

In the Real World

Imagine a drone delivering packages for a service like Zepto. The drone's battery consumption might follow a parabolic curve depending on its speed. Finding the minimum point of this curve helps engineers determine the most energy-efficient speed for the drone to fly, maximizing its range and delivery time.

Key Vocabulary

Key Terms

PARABOLA: A U-shaped curve, the graph of a quadratic equation | VERTEX: The turning point of a parabola, which can be its minimum or maximum point | QUADRATIC EQUATION: An equation of the form y = ax^2 + bx + c, where a is not zero | COEFFICIENTS: The numerical values (a, b, c) in a quadratic equation

What's Next

What to Learn Next

Now that you understand the minimum point, you're ready to explore the 'Maximum Point of a Parabola'. This will show you how to find the highest value a parabola can reach, which is super useful for solving optimization problems in real life!