One More Transformation for the Road (Maybe!)

Okay, okay, I might have fibbed a little...there might be even more transformations down the line. But for now, let's focus on stretching and compressing transformations. These transformations affect the shape of the graph by either pulling it wider or squeezing it narrower.

Stretching and Compressing: The Key Concepts

Remember the general form of a transformed quadratic function: $f(x) = a(x - h)^2 + k$. The value of 'a' plays a crucial role in vertical stretches and compressions:

  • Vertical Stretch: If $|a| > 1$, the graph is stretched vertically. This makes the parabola narrower. For example, in $f(x) = 5x^2$, the graph of $y = x^2$ is stretched by a factor of 5.
  • Vertical Compression: If $0 < |a| < 1$, the graph is compressed vertically. This makes the parabola wider. For example, in $f(x) = \frac{1}{2}x^2$, the graph of $y = x^2$ is compressed by a factor of $\frac{1}{2}$.
  • Reflection: If $a < 0$, the graph is reflected over the x-axis. This flips the parabola upside down. So, $f(x) = -x^2$ is a reflection of the parent function $f(x) = x^2$.

Examples

Let's look at some examples to solidify these concepts:

  • $f(x) = (2x)^2 = 4x^2$: This represents a vertical stretch by a factor of 4. Notice how the '2' inside the parentheses affects the x-values, effectively compressing the graph horizontally, which visually appears as a vertical stretch. The vertex remains at (0,0).
  • $f(x) = \frac{2}{3}x^2$: This represents a vertical compression by a factor of $\frac{2}{3}$ and also a reflection since the coefficient is actually negative. The vertex is still at (0, 0).
  • $f(x) = 5(x - 2)^2 - 3$: This combines several transformations. It's stretched vertically by a factor of 5, shifted 2 units to the right, and shifted 3 units down. The vertex is at (2, -3).

Why Does This Work?

Think about what 'a' does to the y-values. If 'a' is greater than 1, it multiplies each y-value by a number larger than 1, making the graph taller (stretched). If 'a' is between 0 and 1, it multiplies each y-value by a number smaller than 1, making the graph shorter (compressed).

Practice Makes Perfect!

Don't just memorize these rules; understand them! Work through the attached practice problems to master stretching and compressing transformations. Remember to always consider the order of operations when applying multiple transformations.

Resources

  • Video to watch for stretching and compressing transformations
  • Class Notes (Attached)
  • Homework (Attached)
  • Practice A Worksheet (Attached)

Keep up the great work! You're getting closer to mastering transformations!