What is a Stretch of a Graph?

S3-SA5-0291

Grade Level:

Class 9

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

Definition

What is it?

A 'stretch' of a graph means making the graph taller or wider without changing its basic shape or position. It's like pulling a rubber band to make it longer or taller, but it still remains a rubber band. This transformation multiplies the y-coordinates or x-coordinates of the original points on the graph by a certain factor.

Simple Example

Quick Example

Imagine you have a graph showing the price of a samosa over time. If the shop owner decides to double the price of every samosa, the graph would stretch vertically upwards. Each original price point on the graph would now be twice as high, making the entire graph look 'taller' or 'stretched vertically'.

Worked Example

Step-by-Step

Let's say we have a simple graph of y = x. We want to apply a vertical stretch by a factor of 2.

Step 1: Understand the original function. For y = x, some points are (0,0), (1,1), (2,2).
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Step 2: Understand a vertical stretch. A vertical stretch by a factor 'k' means you multiply all the y-coordinates by 'k'. So, y_new = k * y_original.
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Step 3: Apply the stretch. Here, k = 2. So, the new function will be y_new = 2 * x.
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Step 4: Find new points. For the original points:
(0,0) becomes (0, 2*0) = (0,0)
(1,1) becomes (1, 2*1) = (1,2)
(2,2) becomes (2, 2*2) = (2,4)
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Step 5: Compare the graphs. The new graph y = 2x will be steeper than y = x, looking like it has been pulled upwards from the x-axis.

Answer: A vertical stretch of y = x by a factor of 2 results in the new graph y = 2x.

Why It Matters

Understanding graph stretches is vital in fields like Physics to model how springs stretch or how signals change in strength. In Data Science, it helps analyze how data patterns scale up or down. Engineers use it to design structures that can handle different loads and stresses.

Common Mistakes

MISTAKE: Confusing vertical stretch with horizontal stretch. | CORRECTION: Vertical stretch affects the y-coordinates (makes graph taller/shorter), while horizontal stretch affects the x-coordinates (makes graph wider/narrower). Remember 'vertical' relates to 'y' and 'horizontal' relates to 'x'.

MISTAKE: Multiplying the entire x-term by the stretch factor for horizontal stretch. | CORRECTION: For a horizontal stretch by factor 'k', you replace 'x' with 'x/k' in the function. For example, to stretch y = x^2 horizontally by 2, it becomes y = (x/2)^2, not y = (2x)^2.

MISTAKE: Applying the stretch factor in the wrong direction (e.g., dividing for stretch when it should be multiplying). | CORRECTION: For vertical stretch by factor 'k', multiply the function output (y) by 'k'. For horizontal stretch by factor 'k', replace 'x' with 'x/k' (which is effectively multiplying the x-coordinate by 'k').

Practice Questions

Try It Yourself

QUESTION: If the graph of y = x^2 is stretched vertically by a factor of 3, what is the new equation? | ANSWER: y = 3x^2

QUESTION: The graph of y = sin(x) is stretched horizontally by a factor of 2. What is the new equation? | ANSWER: y = sin(x/2)

QUESTION: A graph is represented by y = 2x + 1. If it undergoes a vertical stretch by a factor of 4, and then a horizontal stretch by a factor of 2, what is the final equation? | ANSWER: First vertical stretch: y = 4(2x + 1) = 8x + 4. Then horizontal stretch: y = 8(x/2) + 4 = 4x + 4. So, the final equation is y = 4x + 4.

MCQ

Quick Quiz

Which transformation will make the graph of y = x^3 appear 'taller'?

Replacing x with x/2

Multiplying the entire function by 0.5

Multiplying the entire function by 4

Adding 2 to the entire function

The Correct Answer Is:

Multiplying the entire function by 4 (y = 4x^3) is a vertical stretch by a factor of 4, making the graph taller. Replacing x with x/2 is a horizontal stretch, multiplying by 0.5 is a vertical compression (making it shorter), and adding 2 shifts the graph upwards.

Real World Connection

In the Real World

Imagine a sound engineer adjusting the 'gain' of an audio signal on a music mixing console. Increasing the gain is like applying a vertical stretch to the sound wave graph, making the sound louder. Similarly, a meteorologist might stretch a temperature graph to highlight extreme changes in weather patterns in a specific region of India.

Key Vocabulary

Key Terms

VERTICAL STRETCH: Making a graph taller by multiplying y-coordinates | HORIZONTAL STRETCH: Making a graph wider by multiplying x-coordinates | TRANSFORMATION: Changing the position, size, or orientation of a graph | STRETCH FACTOR: The number by which coordinates are multiplied to stretch a graph

What's Next

What to Learn Next

Great job understanding graph stretches! Next, you should explore 'Graph Compressions' and 'Graph Reflections'. These concepts, along with stretches, are fundamental building blocks for understanding how functions can be transformed and manipulated.