What is a Quadratic Function? | Simple Explanation for Class 10

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What is a Quadratic Function in Real Life?

Grade Level:

Class 10

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

Definition

What is it?

A quadratic function is a mathematical rule that describes how one quantity changes in relation to another, where the highest power of the variable is 2. In real life, it helps us model situations where things go up and then come down, or vice versa, creating a curved path.

Simple Example

Quick Example

Imagine a cricketer hitting a six! The path the cricket ball takes from the bat, flying high in the air, and then landing in the stands, is a curve. This curved path can be described by a quadratic function, showing how the ball's height changes over its horizontal distance.

Worked Example

Step-by-Step

Let's say the height (h) of a ball thrown upwards after 't' seconds is given by the function h(t) = -5t^2 + 20t + 1.

1. **Understand the function:** This is a quadratic function because the highest power of 't' is 2 (t^2).
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2. **Find height at t=0 seconds:** Substitute t=0 into the function.
h(0) = -5(0)^2 + 20(0) + 1 = 0 + 0 + 1 = 1 meter.
This means the ball starts at 1 meter above the ground.
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3. **Find height at t=1 second:** Substitute t=1 into the function.
h(1) = -5(1)^2 + 20(1) + 1 = -5 + 20 + 1 = 16 meters.
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4. **Find height at t=2 seconds:** Substitute t=2 into the function.
h(2) = -5(2)^2 + 20(2) + 1 = -5(4) + 40 + 1 = -20 + 40 + 1 = 21 meters.
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5. **Find height at t=3 seconds:** Substitute t=3 into the function.
h(3) = -5(3)^2 + 20(3) + 1 = -5(9) + 60 + 1 = -45 + 60 + 1 = 16 meters.
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6. **Find height at t=4 seconds:** Substitute t=4 into the function.
h(4) = -5(4)^2 + 20(4) + 1 = -5(16) + 80 + 1 = -80 + 80 + 1 = 1 meter.

ANSWER: The ball's height changes from 1m, to 16m, to 21m, then back to 16m and 1m at different times, showing a curved path.

Why It Matters

Quadratic functions are super important in fields like engineering to design bridges, in physics to understand projectile motion, and even in economics to model profit and loss. Learning them now helps you understand how things work around you and opens doors to careers in AI/ML, data science, and more.

Common Mistakes

MISTAKE: Confusing a quadratic function with a linear function, thinking its graph is a straight line. | CORRECTION: Remember that a quadratic function always has a term with x^2, which makes its graph a U-shaped curve called a parabola.

MISTAKE: Incorrectly identifying the highest power of the variable. For example, in y = 2x + 3x^2, students might think it's not quadratic because x^2 is not first. | CORRECTION: Always look for the term with the highest power of the variable. If it's 2, it's quadratic, regardless of its position in the equation.

MISTAKE: Not understanding that the 'a' coefficient in ax^2 + bx + c = 0 determines the direction of the parabola. | CORRECTION: If 'a' is positive, the parabola opens upwards (like a 'U'). If 'a' is negative, it opens downwards (like an 'n').

Practice Questions

Try It Yourself

QUESTION: Is the function f(x) = 3x^2 - 5x + 7 a quadratic function? Why or why not? | ANSWER: Yes, it is a quadratic function because the highest power of the variable 'x' is 2.

QUESTION: A company's daily profit P (in rupees) is modelled by P(x) = -2x^2 + 80x - 300, where 'x' is the number of items sold. Is this a quadratic function? If 'x' is 10 items, what is the profit? | ANSWER: Yes, it is a quadratic function. For x=10, P(10) = -2(10)^2 + 80(10) - 300 = -2(100) + 800 - 300 = -200 + 800 - 300 = 300 rupees.

QUESTION: The height of a ball thrown is given by h(t) = -t^2 + 6t. At what time(s) will the ball be at a height of 8 units? (Hint: Set h(t) = 8 and solve). | ANSWER: Set -t^2 + 6t = 8. Rearrange to t^2 - 6t + 8 = 0. Factorise to (t-2)(t-4) = 0. So, t = 2 seconds and t = 4 seconds. The ball is at a height of 8 units at 2 seconds and 4 seconds.

MCQ

Quick Quiz

Which of the following real-life scenarios can typically be modelled by a quadratic function?

The cost of mobile data based on how many GB you use (a fixed price per GB).

The straight-line distance an auto-rickshaw travels at a constant speed.

The path of water sprayed from a garden hose.

The total amount of chai sold increasing steadily each hour at a stall.

The Correct Answer Is:

The path of water sprayed from a garden hose forms a parabolic arc, which is the characteristic shape of a quadratic function's graph. Options A, B, and D usually represent linear relationships or constant rates.

Real World Connection

In the Real World

From designing the curved arch of a bridge in Mumbai to predicting the trajectory of a rocket launched by ISRO, quadratic functions are essential. Even in sports analytics, like predicting a cricket ball's landing spot, these functions are used by data scientists to make sense of movement and optimize performance.

Key Vocabulary

Key Terms

QUADRATIC FUNCTION: A function where the highest power of the variable is 2, usually written as ax^2 + bx + c. | PARABOLA: The U-shaped curve that is the graph of a quadratic function. | COEFFICIENT: The number multiplied by a variable in an algebraic term, like 'a' in ax^2. | VERTEX: The highest or lowest point on a parabola. | ROOTS/ZEROS: The x-values where the parabola crosses the x-axis (where the function's value is zero).

What's Next

What to Learn Next

Great job understanding quadratic functions! Next, you should explore how to solve quadratic equations using methods like factoring, completing the square, or the quadratic formula. This will help you find specific points, like when a thrown ball hits the ground!