What is the Relationship Between Coefficients and Vertex?

S3-SA1-0456

Grade Level:

Class 8

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

Definition

What is it?

The relationship between coefficients and the vertex of a parabola tells us how the numbers in a quadratic equation (ax^2 + bx + c) directly determine the turning point of its graph. The vertex is the highest or lowest point of the parabola, and its coordinates can be found using simple formulas derived from the coefficients.

Simple Example

Quick Example

Imagine a cricket ball hit high in the air. Its path forms a parabola. The quadratic equation describes this path, and the vertex tells us the exact point where the ball reaches its maximum height before coming down. The 'a', 'b', and 'c' values in the equation help us find this highest point.

Worked Example

Step-by-Step

Let's find the vertex of the parabola for the equation y = 2x^2 - 8x + 5.

1. Identify the coefficients: Here, a = 2, b = -8, and c = 5.
---2. Use the formula for the x-coordinate of the vertex: x = -b / (2a).
---3. Substitute the values: x = -(-8) / (2 * 2) = 8 / 4 = 2.
---4. Now, substitute this x-value back into the original equation to find the y-coordinate of the vertex: y = 2(2)^2 - 8(2) + 5.
---5. Calculate: y = 2(4) - 16 + 5 = 8 - 16 + 5 = -8 + 5 = -3.
---6. So, the vertex coordinates are (2, -3).

Answer: The vertex of the parabola y = 2x^2 - 8x + 5 is (2, -3).

Why It Matters

Understanding the vertex is crucial in fields like physics to calculate the maximum height of a projectile or in engineering to design stable structures. Data scientists use it to find optimal points in models, and it's even used in AI/ML to optimize learning algorithms, helping computers make smarter decisions.

Common Mistakes

MISTAKE: Forgetting the negative sign in the x-coordinate formula, calculating x = b / (2a) instead of x = -b / (2a). | CORRECTION: Always remember the formula is x = -b / (2a). Pay close attention to negative signs, especially if 'b' itself is negative.

MISTAKE: Substituting the x-coordinate back into the vertex formula (x = -b / (2a)) again instead of the original quadratic equation to find the y-coordinate. | CORRECTION: After finding 'x' for the vertex, substitute this 'x' value into the original quadratic equation (y = ax^2 + bx + c) to get the corresponding 'y' value.

MISTAKE: Incorrectly identifying the coefficients 'a', 'b', and 'c' when the equation is not in standard form (e.g., 3x + 2x^2 - 1 = y). | CORRECTION: Always rearrange the quadratic equation into the standard form y = ax^2 + bx + c first, before identifying 'a', 'b', and 'c'.

Practice Questions

Try It Yourself

QUESTION: Find the x-coordinate of the vertex for the parabola y = x^2 + 6x + 7. | ANSWER: -3

QUESTION: Determine the y-coordinate of the vertex for the parabola y = -x^2 + 4x - 1. | ANSWER: 3

QUESTION: A mobile app's daily users (in thousands) can be modeled by U(t) = -t^2 + 10t + 20, where 't' is the number of weeks since launch. Find the week 't' when the app reaches its maximum users, and how many users (in thousands) it has then. | ANSWER: Week 5, 45 thousand users

MCQ

Quick Quiz

For the quadratic equation y = -3x^2 + 12x - 5, what is the x-coordinate of its vertex?

-2

-1

The Correct Answer Is:

Using the formula x = -b / (2a), with a = -3 and b = 12, we get x = -12 / (2 * -3) = -12 / -6 = 2. So, option B is correct.

Real World Connection

In the Real World

In India, civil engineers use quadratic equations to design bridges and arches. The vertex helps them find the strongest point or the maximum height of an arch for stability. Even in sports analytics for Kabaddi, data scientists might use this to predict a raider's optimal jumping height or trajectory.

Key Vocabulary

Key Terms

COEFFICIENTS: The numerical values (a, b, c) in a quadratic equation (ax^2 + bx + c) | VERTEX: The highest or lowest point on the graph of a parabola | PARABOLA: The U-shaped curve formed by the graph of a quadratic equation | QUADRATIC EQUATION: An equation where the highest power of the variable is 2 (e.g., x^2) | STANDARD FORM: The general way to write a quadratic equation: y = ax^2 + bx + c

What's Next

What to Learn Next

Great job understanding the vertex! Next, you can explore 'How to Graph a Parabola using the Vertex and Intercepts'. This will show you how to draw the entire shape of the parabola accurately, building on what you've learned about its turning point.