What is the Significance of k in y = a(x-h)² + k?

S3-SA5-0207

Grade Level:

Class 10

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

Definition

What is it?

In the quadratic equation y = a(x-h)² + k, the value 'k' represents the vertical shift of the parabola from the x-axis. It tells us the minimum or maximum y-value the parabola reaches, which is the y-coordinate of its vertex. Essentially, 'k' moves the entire graph up or down without changing its shape.

Simple Example

Quick Example

Imagine you're tracking the daily sales of 'chai' at a stall. If the base sales are 'y = a(x-h)²', and then the owner decides to add a special offer that guarantees 50 extra cups sold every day, regardless of other factors, this '50' is like 'k'. The entire sales graph shifts up by 50 cups.

Worked Example

Step-by-Step

Let's find the significance of 'k' in the equation y = 2(x-3)² + 5.

1. Identify the standard form: The given equation is y = 2(x-3)² + 5, which matches the form y = a(x-h)² + k.
2. Identify the value of 'k': In this equation, k = 5.
3. Interpret 'k': Since k = 5, this means the parabola is shifted 5 units upwards from the x-axis.
4. Find the vertex: The vertex of the parabola is at (h, k). Here, h = 3 and k = 5. So, the vertex is (3, 5).
5. Significance: The y-coordinate of the vertex is 5. This tells us the lowest point (minimum value) of the parabola is at y = 5. This is because 'a' (which is 2) is positive, so the parabola opens upwards.

Answer: The value k = 5 signifies that the parabola's vertex has a y-coordinate of 5, meaning the entire graph is shifted 5 units up, and its minimum value is 5.

Why It Matters

Understanding 'k' helps in predicting outcomes and optimizing systems in fields like AI/ML and Data Science. Engineers use it to design structures that can withstand specific loads, and economists use it to model market trends and predict peak performance or lowest costs. It's crucial for understanding how a base value is affected by constant additions or subtractions.

Common Mistakes

MISTAKE: Confusing 'k' with the x-coordinate of the vertex. | CORRECTION: 'k' is always the y-coordinate of the vertex. Remember, 'h' is for the horizontal shift (x-coordinate), and 'k' is for the vertical shift (y-coordinate).

MISTAKE: Assuming 'k' always represents a minimum value. | CORRECTION: 'k' represents a minimum value only if 'a' is positive (parabola opens upwards). If 'a' is negative (parabola opens downwards), 'k' represents the maximum value.

MISTAKE: Forgetting that 'k' affects the entire graph vertically. | CORRECTION: 'k' shifts every point on the parabola up or down by the same amount. It's not just the vertex that moves; the whole graph translates.

Practice Questions

Try It Yourself

QUESTION: For the equation y = 3(x+1)² - 7, what is the value of 'k' and what does it represent? | ANSWER: k = -7. It represents a vertical shift of 7 units downwards, and the minimum y-value of the parabola is -7.

QUESTION: If a parabola has its vertex at (4, 10) and opens downwards, what is the value of 'k' in its equation? | ANSWER: k = 10.

QUESTION: A company's profit (P) for a product can be modeled by P = -2(x-50)² + 1000, where 'x' is the number of items sold. What is the maximum profit the company can make, and which part of the equation tells you this? | ANSWER: The maximum profit is 1000. The value 'k' (which is 1000) tells us this, because 'a' is negative, meaning the parabola opens downwards and 'k' is the maximum point.

MCQ

Quick Quiz

In the equation y = -5(x-2)² + 8, what is the significance of the value 8?

It is the x-coordinate of the vertex.

It is the maximum y-value of the parabola.

It is the minimum y-value of the parabola.

It shifts the parabola 8 units to the left.

The Correct Answer Is:

The value 'k' is 8. Since 'a' is -5 (negative), the parabola opens downwards, so 'k' represents the maximum y-value. Options A and D are incorrect as 'h' handles horizontal shifts and x-coordinate. Option C is wrong because 'a' is negative.

Real World Connection

In the Real World

Imagine an engineer at ISRO designing a satellite's trajectory. If the base path is a parabola, 'k' could represent the initial altitude offset from a reference point. Or, in cricket analytics, if a batsman's scoring rate follows a parabolic curve, 'k' could be the base runs added due to a 'free hit' or a fixed bonus, shifting their entire score graph upwards.

Key Vocabulary

Key Terms

VERTEX: The turning point of a parabola, either its highest or lowest point. | PARABOLA: The U-shaped curve that is the graph of a quadratic equation. | VERTICAL SHIFT: Moving a graph up or down on the coordinate plane. | QUADRATIC EQUATION: An equation of degree 2, often written as y = ax² + bx + c or y = a(x-h)² + k.

What's Next

What to Learn Next

Great job understanding 'k'! Next, explore the significance of 'h' in y = a(x-h)² + k. Understanding 'h' will complete your knowledge of how a parabola's vertex is determined and how its position changes on the graph.