What is the Y-intercept of a Parabola?

S3-SA1-0190

Grade Level:

Class 7

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

Definition

What is it?

The Y-intercept of a parabola is the point where the parabola crosses or touches the Y-axis. At this point, the X-coordinate is always zero. It tells us the 'starting value' or the value of Y when X is zero.

Simple Example

Quick Example

Imagine you are throwing a cricket ball, and its path forms a parabola. The Y-intercept would be the height of the ball when it leaves your hand (at time X=0). If you release the ball from a height of 1.5 meters, then the Y-intercept is 1.5.

Worked Example

Step-by-Step

Let's find the Y-intercept for the parabola given by the equation: Y = 2X^2 - 3X + 5.

Step 1: Remember that at the Y-intercept, the X-coordinate is always 0.

---Step 2: Substitute X = 0 into the equation.
Y = 2(0)^2 - 3(0) + 5

---Step 3: Simplify the terms with X.
Y = 2(0) - 0 + 5

---Step 4: Continue simplifying.
Y = 0 - 0 + 5

---Step 5: Calculate the final value of Y.
Y = 5

---Answer: The Y-intercept is 5. This means the parabola crosses the Y-axis at the point (0, 5).

Why It Matters

Understanding the Y-intercept helps scientists and engineers understand starting conditions in many real-world problems. In Data Science, it can represent the initial cost or value. In Physics, it might show the initial height of a projectile, which is crucial for predicting its path.

Common Mistakes

MISTAKE: Confusing the Y-intercept with the X-intercepts. | CORRECTION: The Y-intercept is where X=0. X-intercepts are where Y=0.

MISTAKE: Substituting Y=0 instead of X=0 into the equation. | CORRECTION: Always substitute X=0 to find the Y-intercept.

MISTAKE: Forgetting that the Y-intercept is a point (0, Y-value) and just stating the Y-value. | CORRECTION: While often we just say the Y-value, remember it's a point on the graph, so (0, Y-value) is the full coordinate.

Practice Questions

Try It Yourself

QUESTION: What is the Y-intercept of the parabola Y = X^2 + 4X + 3? | ANSWER: 3 (or (0, 3))

QUESTION: Find the Y-intercept of the parabola Y = -3X^2 + 7X - 1. | ANSWER: -1 (or (0, -1))

QUESTION: A parabola has the equation Y = (X - 2)(X + 5). What is its Y-intercept? (Hint: First expand the equation). | ANSWER: -10 (or (0, -10))

MCQ

Quick Quiz

For the equation Y = 5X^2 - 2X + 8, what is the Y-intercept?

-2

The Correct Answer Is:

To find the Y-intercept, we set X=0 in the equation. So, Y = 5(0)^2 - 2(0) + 8, which simplifies to Y = 8. Therefore, the correct answer is 8.

Real World Connection

In the Real World

Imagine a drone delivering a package for a service like Zomato or Swiggy. If its flight path is modelled by a parabola, the Y-intercept would tell us the initial height from which the drone takes off. This is important for calculating fuel, flight time, and avoiding obstacles.

Key Vocabulary

Key Terms

PARABOLA: A U-shaped curve that is the graph of a quadratic equation. | Y-AXIS: The vertical line in a coordinate plane. | X-AXIS: The horizontal line in a coordinate plane. | COORDINATE: A set of numbers that show an exact position on a graph. | QUADRATIC EQUATION: An equation where the highest power of the variable is 2.

What's Next

What to Learn Next

Now that you understand the Y-intercept, you can explore X-intercepts (also called roots or zeroes) of a parabola. These points tell us where the parabola crosses the X-axis and are often found by solving quadratic equations, which is a very important skill!