What is the Significance of c in y = ax^2 + bx + c? | Simple Explanation for Class 10

S3-SA5-0205

What is the Significance of c in y = ax² + bx + c?

Grade Level:

Class 10

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

Definition

What is it?

In the quadratic equation y = ax^2 + bx + c, 'c' is the constant term. It represents the y-intercept of the parabola, which is the point where the graph of the quadratic equation crosses the y-axis.

Simple Example

Quick Example

Imagine a cricket ball hit high in the air. Its path can be modeled by a quadratic equation. The 'c' value would tell us the initial height of the ball when it was hit, before it started its upward or downward journey.

Worked Example

Step-by-Step

PROBLEM: Find the significance of 'c' in the equation y = 2x^2 - 4x + 5.
---
STEP 1: Understand that 'c' is the constant term in the quadratic equation y = ax^2 + bx + c.
---
STEP 2: In the given equation, y = 2x^2 - 4x + 5, compare it to the standard form.
---
STEP 3: Identify the value of 'c'. Here, c = 5.
---
STEP 4: Recall that 'c' represents the y-intercept. This means when x = 0, y = c.
---
STEP 5: Substitute x = 0 into the equation: y = 2(0)^2 - 4(0) + 5.
---
STEP 6: Calculate y: y = 0 - 0 + 5 = 5.
---
ANSWER: The significance of 'c' is that it tells us the y-intercept of the graph is 5. So, the parabola crosses the y-axis at the point (0, 5).

Why It Matters

Understanding 'c' helps in AI/ML to set initial conditions for models and in Physics to determine starting positions or initial energy. Engineers use it to design structures like bridges, knowing the base height. This concept is fundamental for careers in data science and engineering.

Common Mistakes

MISTAKE: Thinking 'c' is always positive. | CORRECTION: 'c' can be any real number, positive, negative, or zero, depending on where the graph crosses the y-axis.

MISTAKE: Confusing 'c' with the x-intercepts (roots). | CORRECTION: 'c' is specifically the y-intercept (where x=0), while x-intercepts are where y=0.

MISTAKE: Not recognizing 'c' when the equation is not in standard form (e.g., y = x^2 + 5x). | CORRECTION: If there's no constant term, it means c = 0. The equation can be written as y = x^2 + 5x + 0.

Practice Questions

Try It Yourself

QUESTION: For the equation y = -x^2 + 3x - 7, what is the value of 'c' and what does it represent? | ANSWER: c = -7. It represents the y-intercept, meaning the graph crosses the y-axis at (0, -7).

QUESTION: A quadratic equation models the profit (y) of a company based on the number of items sold (x). If the equation is y = -0.5x^2 + 10x - 50, what does the 'c' value mean in this business context? | ANSWER: c = -50. It means that if 0 items are sold (x=0), the company has a loss of 50 rupees (initial cost or fixed overhead).

QUESTION: If a parabola passes through the point (0, 8), what can you say about the 'c' value of its quadratic equation y = ax^2 + bx + c? | ANSWER: Since the point (0, 8) is the y-intercept, the value of 'c' must be 8.

MCQ

Quick Quiz

What is the significance of the constant 'c' in the quadratic equation y = ax^2 + bx + c?

It determines the width of the parabola.

It is the x-intercept of the parabola.

It is the y-intercept of the parabola.

It shifts the parabola horizontally.

The Correct Answer Is:

The constant 'c' in a quadratic equation represents the y-intercept, which is the point where the graph crosses the y-axis (when x=0). Options A, B, and D describe other properties or parameters.

Real World Connection

In the Real World

In a logistics company like Zepto, a quadratic equation might model the fuel efficiency (y) of a delivery van based on its speed (x). The 'c' value could represent the fuel consumed even when the van is stationary (idling), which is an important factor for operational costs.

Key Vocabulary

Key Terms

QUADRATIC EQUATION: An equation of the form y = ax^2 + bx + c where a is not zero. | Y-INTERCEPT: The point where a graph crosses the y-axis. | CONSTANT TERM: The term in an algebraic expression that has a fixed value and does not contain any variables. | PARABOLA: The U-shaped curve that is the graph of a quadratic equation.

What's Next

What to Learn Next

Next, explore the significance of 'a' and 'b' in the quadratic equation. Understanding 'a' will tell you about the parabola's opening direction and width, while 'b' influences the position of its vertex. Together, 'a', 'b', and 'c' paint a complete picture!