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What is the Vertex of a Parabola (-b/2a)?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The vertex of a parabola is the highest or lowest point on its U-shaped curve. It's the turning point where the parabola changes direction. For a parabola given by the equation y = ax^2 + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / (2a).
Simple Example
Quick Example
Imagine you throw a cricket ball up in the air. The path it takes is a parabola. The highest point the ball reaches before it starts coming down is the vertex. This point shows the maximum height the ball achieved.
Worked Example
Step-by-Step
Let's find the x-coordinate of the vertex for the parabola y = 2x^2 - 8x + 5.
Step 1: Identify the values of 'a' and 'b' from the equation.
Here, a = 2 and b = -8.
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Step 2: Use the formula for the x-coordinate of the vertex: x = -b / (2a).
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Step 3: Substitute the values of 'a' and 'b' into the formula.
x = -(-8) / (2 * 2)
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Step 4: Simplify the expression.
x = 8 / 4
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Step 5: Calculate the final value.
x = 2
The x-coordinate of the vertex for this parabola is 2.
Why It Matters
Understanding the vertex helps scientists predict the path of rockets or how far a javelin will travel. In AI and Data Science, finding the minimum or maximum point (like a vertex) helps algorithms make better decisions. Engineers use it to design bridges and antennas, ensuring they are stable and efficient.
Common Mistakes
MISTAKE: Forgetting the negative sign in front of 'b' in the formula. For example, if b is -8, writing -8 / (2a) instead of -(-8) / (2a). | CORRECTION: Always remember the formula is x = -b / (2a). If 'b' is already negative, then -(-b) will become positive.
MISTAKE: Mixing up 'a' and 'b' values, or using 'c' in the vertex formula. | CORRECTION: Remember 'a' is the number with x^2, 'b' is the number with x, and 'c' is the constant. The formula for the x-coordinate of the vertex only uses 'a' and 'b'.
MISTAKE: Not multiplying 'a' by 2 in the denominator, or doing (2+a) instead of (2*a). | CORRECTION: The formula is 2a, which means 2 multiplied by 'a'. Always perform the multiplication correctly in the denominator first.
Practice Questions
Try It Yourself
QUESTION: Find the x-coordinate of the vertex for the parabola y = x^2 - 6x + 10. | ANSWER: 3
QUESTION: A parabola is given by the equation y = -3x^2 + 12x - 7. What is the x-coordinate of its vertex? | ANSWER: 2
QUESTION: If the x-coordinate of the vertex of a parabola y = ax^2 + 10x + 1 is 5, what is the value of 'a'? | ANSWER: -1
MCQ
Quick Quiz
Which formula gives the x-coordinate of the vertex for a parabola y = ax^2 + bx + c?
x = b / (2a)
x = -b / (2a)
x = -c / (2a)
x = -b / a
The Correct Answer Is:
B
The correct formula for the x-coordinate of the vertex is x = -b / (2a). Option B correctly shows this formula. The other options are incorrect variations.
Real World Connection
In the Real World
When ISRO launches a satellite, engineers use parabolic paths and their vertices to calculate the optimal trajectory for the rocket to reach orbit. Similarly, in cricket, analysts use parabolas to predict the flight path of a shot, and the vertex tells them the maximum height the ball will reach, which helps in field placement.
Key Vocabulary
Key Terms
PARABOLA: A U-shaped curve that is the graph of a quadratic equation. | VERTEX: The turning point of a parabola, either its highest or lowest point. | QUADRATIC EQUATION: An equation where the highest power of the variable is 2 (like x^2). | COEFFICIENT: A number multiplied by a variable in an algebraic term (e.g., 'a' in ax^2).
What's Next
What to Learn Next
Now that you know how to find the x-coordinate of the vertex, you can learn how to find the y-coordinate as well! This will give you the complete coordinates of the vertex, which is super useful for graphing parabolas.
