What is the Minimum Value of a Quadratic Function?

S6-SA1-0128

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The minimum value of a quadratic function is the lowest point its graph (a parabola) can reach. For a quadratic function written as ax^2 + bx + c, this minimum value occurs when the parabola opens upwards (meaning 'a' is positive).

Simple Example

Quick Example

Imagine you throw a cricket ball straight up in the air. Its path forms a curve. The highest point the ball reaches before coming down is like the maximum value. If the curve was upside down, like a valley, the lowest point would be the minimum value. We are finding that lowest point for a specific type of curve.

Worked Example

Step-by-Step

Let's find the minimum value of the quadratic function y = x^2 - 4x + 7.

1. Identify 'a', 'b', and 'c': In y = x^2 - 4x + 7, a = 1, b = -4, c = 7.
---2. Check if 'a' is positive: Here, a = 1, which is positive. So, the parabola opens upwards and has a minimum value.
---3. Find the x-coordinate of the vertex: Use the formula x = -b / (2a). x = -(-4) / (2 * 1) = 4 / 2 = 2.
---4. Substitute this x-value back into the original function to find the minimum y-value: y = (2)^2 - 4(2) + 7.
---5. Calculate the value: y = 4 - 8 + 7.
---6. Simplify: y = -4 + 7 = 3.

The minimum value of the function is 3.

Why It Matters

Understanding minimum values helps engineers design bridges to withstand minimum stress, and helps scientists optimize drug dosages in medicine. In AI/ML, it's used to find the 'best fit' model by minimizing errors, which is crucial for making smart apps like facial recognition or language translation work.

Common Mistakes

MISTAKE: Assuming every quadratic function has a minimum value. | CORRECTION: Only quadratic functions where 'a' (the coefficient of x^2) is positive have a minimum value. If 'a' is negative, it has a maximum value.

MISTAKE: Mixing up the formula for the x-coordinate of the vertex. | CORRECTION: The x-coordinate of the vertex is always given by x = -b / (2a). Remember the negative sign!

MISTAKE: Forgetting to substitute the x-coordinate back into the ORIGINAL function. | CORRECTION: After finding the x-coordinate of the vertex, always plug it into y = ax^2 + bx + c to get the actual minimum y-value.

Practice Questions

Try It Yourself

QUESTION: Find the minimum value of the quadratic function y = x^2 + 6x + 5. | ANSWER: -4

QUESTION: A mobile app's daily profit (in thousands of rupees) can be modeled by P(x) = x^2 - 10x + 30, where x is the number of ads displayed. What is the minimum profit the app can make? | ANSWER: 5 thousand rupees

QUESTION: For the quadratic function f(x) = 2x^2 - 12x + 15, determine if it has a minimum or maximum value, and then find that value. | ANSWER: It has a minimum value of -3.

MCQ

Quick Quiz

Which of the following quadratic functions has a minimum value of -2?

y = x^2 + 4x + 2

y = x^2 - 4x + 2

y = x^2 - 2x + 1

y = x^2 + 2x + 1

The Correct Answer Is:

For y = x^2 + 4x + 2, the x-coordinate of the vertex is -4/(2*1) = -2. Substituting x = -2 into the function gives y = (-2)^2 + 4(-2) + 2 = 4 - 8 + 2 = -2. Options B, C, and D yield different minimum values.

Real World Connection

In the Real World

Imagine a drone delivering a package for a company like Zepto. Its flight path might be modeled by a quadratic function. Finding the minimum value helps engineers calculate the lowest point the drone will reach, ensuring it doesn't hit obstacles like trees or buildings, making deliveries safe and efficient.

Key Vocabulary

Key Terms

QUADRATIC FUNCTION: A function of the form ax^2 + bx + c, where a, b, c are constants and a is not zero. | PARABOLA: The U-shaped curve that is the graph of a quadratic function. | VERTEX: The highest or lowest point on a parabola. | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., 'a' in ax^2).

What's Next

What to Learn Next

Great job understanding minimum values! Next, explore 'Maximum Value of a Quadratic Function'. It uses a similar approach but focuses on parabolas that open downwards, helping you grasp the full picture of quadratic function behavior.