2.9.2 Stretches

Transformations of functions are ways to modify a function's graph by shifting, stretching, compressing, or reflecting it. Stretches are a specific type of transformation where the graph of a function is expanded or contracted either vertically or horizontally.

Types of Stretches

1. Vertical Stretch:

• A vertical stretch involves multiplying the entire function by a constant factor.
• If $\ y = f(x)$, then $\ y = a \cdot f(x)$ represents a vertical stretch if $\ |a| > 1$ or a vertical compression if $\ 0 < |a| < 1 .$
• Effect: The graph is stretched vertically by a factor of $\ a$. Points on the graph move further away from the $x$-axis if $\ |a| > 1$ or closer if $\ 0 < |a| < 1$.

2. Horizontal Stretch:

• A horizontal stretch involves multiplying the input $\ x$ by a constant factor.
• If $\ y = f(x)$, then $y = f\left(\frac{x}{b}\right)$ represents a horizontal stretch if $\ |b| > 1$ or a horizontal compression if $\ 0 < |b| < 1 .$
• Effect: The graph is stretched horizontally by a factor of $\ b$. Points on the graph move away from the $y$-axis if $\ |b| > 1$ or closer if$\ 0 < |b| < 1$.

Summary of Stretches

• Vertical Stretch by $\ a : \ y = af(x)$
  • Stretches the graph vertically by a factor of $\ |a|$.
  • If $\ a > 1$, the graph is stretched; if $\ 0 < a < 1 ,$ it is compressed.
• Horizontal Stretch by$\ b : \ y = f\left(\frac{x}{b}\right)$
  • Stretches the graph horizontally by a factor of $\ |b|$.
  • If $\ b > 1$, the graph is stretched; if $\ 0 < b < 1$, it is compressed.

Practice Question:

Solution:

3. For $\ y = 3\sin(x) :$

• This is a vertical stretch by a factor of 3.
• The amplitude of the sine wave increases from 1 to 3.

4. For $\ y = \sin\left(\frac{x}{2}\right)$:

• This is a horizontal stretch by a factor of 2.
• The period of the sine wave increases, so the wave repeats every $\ 4\pi\$instead of $\ 2\pi .$

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