What is the Graph of y = a(x-h)² + k?

S3-SA5-0202

Grade Level:

Class 10

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

Definition

What is it?

The graph of y = a(x-h)² + k is a 'U' shaped curve called a parabola. This equation is the 'vertex form' of a quadratic function, where (h, k) represents the lowest or highest point of the parabola, called the vertex.

Simple Example

Quick Example

Imagine you are throwing a cricket ball. The path the ball takes through the air is a parabola. If we plot its height (y) against its horizontal distance (x), the equation y = a(x-h)² + k can describe this path, with (h, k) being the ball's maximum height.

Worked Example

Step-by-Step

Let's sketch the graph of y = 2(x-3)² + 1.
1. Identify the vertex (h, k). Here, h = 3 and k = 1. So, the vertex is (3, 1).
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2. Since 'a' is 2 (a > 0), the parabola opens upwards.
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3. Find a few points. Let x = 2: y = 2(2-3)² + 1 = 2(-1)² + 1 = 2(1) + 1 = 3. So, (2, 3) is a point.
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4. Let x = 4: y = 2(4-3)² + 1 = 2(1)² + 1 = 2(1) + 1 = 3. So, (4, 3) is a point (parabolas are symmetric).
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5. Let x = 1: y = 2(1-3)² + 1 = 2(-2)² + 1 = 2(4) + 1 = 8 + 1 = 9. So, (1, 9) is a point.
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6. Let x = 5: y = 2(5-3)² + 1 = 2(2)² + 1 = 2(4) + 1 = 8 + 1 = 9. So, (5, 9) is a point.
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7. Plot these points (3,1), (2,3), (4,3), (1,9), (5,9) and draw a smooth U-shaped curve through them.
Answer: The graph is an upward-opening parabola with its vertex at (3, 1).

Why It Matters

Understanding parabolas is key in fields like AI/ML for optimizing models and in Physics for studying projectile motion. Engineers use them to design bridges and satellite dishes, while economists might use them to model supply and demand curves.

Common Mistakes

MISTAKE: Confusing the sign of 'h' in (x-h). For y = (x+3)^2 + 1, students might say h=3. | CORRECTION: The form is (x-h). So, (x+3) means (x - (-3)), making h = -3. The vertex is (-3, 1).

MISTAKE: Incorrectly determining if the parabola opens up or down. Students might think 'a' refers to the y-intercept. | CORRECTION: If 'a' is positive (a > 0), the parabola opens upwards. If 'a' is negative (a < 0), it opens downwards.

MISTAKE: Mixing up the x and y coordinates of the vertex. | CORRECTION: The vertex is always (h, k), where 'h' is the x-coordinate and 'k' is the y-coordinate.

Practice Questions

Try It Yourself

QUESTION: What is the vertex of the parabola y = 3(x-5)^2 + 2? | ANSWER: (5, 2)

QUESTION: Does the graph of y = -0.5(x+1)^2 - 4 open upwards or downwards? | ANSWER: Downwards (because a = -0.5, which is negative).

QUESTION: Sketch the graph of y = (x-2)^2 + 3 by finding its vertex and two other points. | ANSWER: Vertex (2, 3). Points could be (1, 4) and (3, 4).

MCQ

Quick Quiz

For the graph of y = -2(x+4)^2 + 5, what is its vertex?

(4, 5)

(-4, 5)

(-4, -5)

(4, -5)

The Correct Answer Is:

The vertex form is y = a(x-h)^2 + k. Here, (x+4) means (x - (-4)), so h = -4. And k = 5. Therefore, the vertex is (-4, 5).

Real World Connection

In the Real World

Think about the dish antennas used for DTH TV or ISRO's satellite communication. These dishes are shaped like parabolas because of their unique property to focus all incoming parallel signals to a single point, which is where the receiver is placed.

Key Vocabulary

Key Terms

Parabola: A U-shaped curve, the graph of a quadratic function. | Vertex: The turning point of a parabola, either its lowest or highest point. | Vertex Form: The equation y = a(x-h)^2 + k, which directly shows the vertex (h, k). | Quadratic Function: A function of the form y = ax^2 + bx + c, whose graph is a parabola.

What's Next

What to Learn Next

Great job understanding the vertex form! Next, explore how to convert a quadratic equation from the standard form (y = ax^2 + bx + c) to the vertex form. This will help you find the vertex of any parabola easily.