Parent Functions & Their Transformations
A **parent function** is the most basic form of a function family. For example, $y=x^2$ is the parent function for all parabolas.
A **transformation** is a change made to a function that alters its graph. The general form of a transformed function is: $$ y = a \cdot f(b(x-h)) + k $$
- **$a$**: **Vertical Stretch/Compression** and **Reflection** across the x-axis. A negative $a$ reflects the graph. If $|a| > 1$, it's a stretch. If $0 < |a| < 1$, it's a compression.
- **$b$**: **Horizontal Stretch/Compression** and **Reflection** across the y-axis. A negative $b$ reflects the graph. It's counter-intuitive: if $|b| > 1$, it's a compression. If $0 < |b| < 1$, it's a stretch.
- **$h$**: **Horizontal Translation** (shift left or right). This is also counter-intuitive: $x-h$ shifts right, while $x+h$ shifts left.
- **$k$**: **Vertical Translation** (shift up or down). This is intuitive: $+k$ shifts up, and $-k$ shifts down.
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Video: The Transformations in Action! 🎬
Watch this video to visualize each transformation applied to a parent function, so you can understand the effect of each parameter. 📈
Practice Problems
Describe the transformations in each function!
1) Describe the transformation of $f(x) = (x+2)^2$.
2) Describe the transformation of $g(x) = -|x|$.
3) Describe the transformation of $h(x) = \sqrt{x} + 3$.
4) Describe the transformation of $p(x) = \frac{1}{2}x^3$.
⚡ Transformations Speed Quiz
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