What is the Turning Point of a Parabola?

S3-SA1-0728

Grade Level:

Class 8

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

Definition

What is it?

The turning point of a parabola is the point where the parabola changes direction. It is either the lowest point (minimum) or the highest point (maximum) on the entire curve. This special point is also called the 'vertex'.

Simple Example

Quick Example

Imagine throwing a cricket ball upwards. It goes up, reaches a maximum height, and then starts coming down. The highest point the ball reaches before falling is the turning point of its path, which forms a parabola.

Worked Example

Step-by-Step

Let's find the turning point (vertex) of the parabola given by the equation y = x^2 - 4x + 3.

Step 1: Identify the coefficients. For a parabola y = ax^2 + bx + c, here a = 1, b = -4, c = 3.
---Step 2: Use the formula for the x-coordinate of the vertex: x = -b / (2a).
---Step 3: Substitute the values: x = -(-4) / (2 * 1) = 4 / 2 = 2.
---Step 4: Now find the y-coordinate by plugging the x-value (x=2) back into the original equation: y = (2)^2 - 4(2) + 3.
---Step 5: Calculate: y = 4 - 8 + 3.
---Step 6: So, y = -1.
---Step 7: The turning point (vertex) is (x, y).
---Answer: The turning point of the parabola y = x^2 - 4x + 3 is (2, -1).

Why It Matters

Understanding turning points is crucial in fields like AI/ML for optimizing models, and in Physics for calculating maximum projectile height. Engineers use it to design bridges and structures for maximum strength, making our lives safer and more efficient.

Common Mistakes

MISTAKE: Confusing the turning point with where the parabola crosses the x-axis (roots). | CORRECTION: The turning point is where the parabola changes direction (maximum/minimum), not necessarily where y=0.

MISTAKE: Forgetting to substitute the x-coordinate back into the original equation to find the y-coordinate of the turning point. | CORRECTION: After finding x = -b/(2a), always plug this x-value into y = ax^2 + bx + c to get the corresponding y-coordinate.

MISTAKE: Making sign errors when using the formula x = -b/(2a), especially with negative 'b' values. | CORRECTION: Be very careful with negative signs. For example, if b = -5, then -b is -(-5) which is +5.

Practice Questions

Try It Yourself

QUESTION: Find the turning point of the parabola y = x^2 - 6x + 5. | ANSWER: (3, -4)

QUESTION: What is the turning point of the parabola y = -x^2 + 2x + 10? (Hint: The parabola opens downwards). | ANSWER: (1, 11)

QUESTION: A company's daily profit (in thousands of rupees) from selling 'x' number of smartwatches is given by P(x) = -x^2 + 10x - 16. Find the number of smartwatches they should sell to get maximum profit, and what that maximum profit is. | ANSWER: Sell 5 smartwatches for a maximum profit of 9 thousand rupees (Rs. 9000).

MCQ

Quick Quiz

Which of the following describes the turning point of a parabola?

The point where the parabola crosses the x-axis

The highest or lowest point on the parabola

The point where the parabola crosses the y-axis

The starting point of the parabola

The Correct Answer Is:

The turning point is the vertex, which represents either the maximum (highest) or minimum (lowest) value of the parabolic function. Options A and C describe intercepts, and D is incorrect as parabolas extend infinitely.

Real World Connection

In the Real World

In India, ISRO scientists use parabolas to design satellite dishes. The turning point helps them find the exact focus where signals are strongest. Also, in cricket analytics, turning points can show the peak performance of a player or team over time, helping coaches strategize.

Key Vocabulary

Key Terms

PARABOLA: A U-shaped curve that is the graph of a quadratic equation | VERTEX: Another name for the turning point of a parabola | QUADRATIC EQUATION: An equation of the form y = ax^2 + bx + c, where a is not zero | MINIMUM: The lowest point on a parabola that opens upwards | MAXIMUM: The highest point on a parabola that opens downwards

What's Next

What to Learn Next

Great job understanding turning points! Next, you can explore 'Quadratic Equations and Their Roots'. This will help you understand where parabolas cross the x-axis, connecting it with the turning point to fully sketch and analyze these important curves.