Stretching and shrinking are important ways to change the shapes and sizes of function graphs. These changes are very useful for students in Grade 12 Algebra I, as they help prepare for more advanced topics like calculus and analytic geometry. Here are some key points about how stretching and shrinking work:

1. Vertical Stretching and Shrinking

• What It Is: Vertical stretching happens when we multiply the output values of a function by a number greater than 1. Vertical shrinking happens when we multiply the output values by a number less than 1 but greater than 0.

• How It Looks Mathematically:

  • If we have a function $f(x)$, a vertical stretch by a factor of $a$ (where $a > 1$) can be shown as $g(x) = a \cdot f(x)$.
  • A vertical shrink by a factor of $b$ (where $0 < b < 1$) is shown as $h(x) = b \cdot f(x)$.

• What Happens to the Graph:

  • A vertical stretch makes the graph points higher and further apart from the x-axis. For example, with $g(x) = 2f(x)$, the points of the graph go twice as high compared to $f(x)$.
  • A vertical shrink makes the graph flatter and brings the points closer to the x-axis. For example, with $h(x) = \frac{1}{2}f(x)$, each point's height is halved.

2. Horizontal Stretching and Shrinking

• What It Is: Horizontal stretching happens when we multiply the input values (the x-values) by a number less than 1. Horizontal shrinking occurs when we multiply the input by a number greater than 1.

• How It Looks Mathematically:

  • For a function $f(x)$, if we do a horizontal stretch by a factor of $b$ (where $b < 1$), we write it as $g(x) = f(bx)$.
  • A horizontal shrink by a factor of $a$ (where $a > 1$) is written as $h(x) = f(\frac{1}{a} x)$.

• What Happens to the Graph:

  • A horizontal stretch makes the graph wider along the x-axis. For example, $g(x) = f(\frac{1}{2} x)$ spreads the points further apart.
  • A horizontal shrink makes the graph narrower along the x-axis. An example is $h(x) = f(2x)$, which pulls the points closer together.

3. Combining Transformations

• Sometimes we mix stretching and shrinking to create different effects on graphs. For instance, we could stretch a graph vertically and shrink it horizontally all at once.

• We can show this with a formula: $g(x) = a \cdot f(bx)$ Here, $a$ is for vertical changes, and $b$ is for horizontal changes.

4. Seeing the Changes

• Using graphing tools helps us see how these transformations affect the graphs. By drawing the original function alongside the changed version, students can understand the visual effects of stretching and shrinking.

• Important details like intercepts (where the graph crosses the axes) and how the graph behaves at its edges may change too. Understanding these changes helps students predict how functions will act in different situations.

Conclusion

In summary, stretching and shrinking are great ways to change function graphs. They not only change how graphs look and their sizes but also how they behave in important ways. Learning these transformations is essential for exploring more about functions and their uses in math. By grasping how these changes work both mathematically and visually, students will improve their analytical skills as they learn more in mathematics.