Transformations of parent functions
Transformations of Parent Functions
Transformations of Parent Functions is a topic in Relations & Functions in the Common Core State Standards. It is aligned to Standard F-BF.3, which requires students to identify the effect on a graph of replacing f(x) by transformations such as f(x)+k, af(x), and f(x-h).
Transformations shift, stretch, and reflect a parent function through \(y=a\,f(b(x-h))+k\) without changing its family.
Theory
A parent function is the simplest form of a family (\(x^2,\ |x|,\ \sqrt{x},\ 2^x,\ \dots\)). Transformations act through:
- \(h\): horizontal shift (right if \(h>0\)).
- \(k\): vertical shift (up if \(k>0\)).
- \(a\): vertical stretch \(|a|\), reflection over the \(x\)-axis if \(a<0\).
- \(b\): horizontal compression by \(|b|\).
The transformation template:
How to apply a transformation
- Identify the parent function.
- Read \(a,\ b,\ h,\ k\) from the equation.
- Apply horizontal changes (\(h,\ b\)) to \(x\), then vertical (\(a,\ k\)) to \(y\).
- Plot the transformed key points.
\(h=4\) inside the square shifts the graph.
| \((x-4)^2\) | \(\Rightarrow\) | \(\text{shift right } 4\) |
The factor \(a=-3\) stretches and reflects.
| \(-3x^2\) | \(\Rightarrow\) | \(\text{stretch } \times 3,\ \text{flip over the } x\text{-axis}\) |
Read \(a=2,\ h=-1,\ k=-5\).
| \(h=-1\) | \(\Rightarrow\) | \(\text{left } 1\) |
| \(a=2\) | \(\Rightarrow\) | \(\text{stretch } \times 2\) |
| \(k=-5\) | \(\Rightarrow\) | \(\text{down } 5\) |
Use \(h=3,\ k=-2\).
| \(y\) | \(=\) | \(|x-3|-2\) |
Common pitfalls
Frequently asked questions
What is a parent function?
The simplest function of a family, such as \(x^2\) for quadratics.
How does \((x-h)\) shift a graph?
It shifts the graph right by \(h\) (inside changes act in the opposite direction).
What does the factor \(a\) do?
It stretches vertically by \(|a|\) and reflects over the \(x\)-axis if \(a<0\).
Do inside or outside changes affect \(y\)?
Outside changes (\(a,\ k\)) affect \(y\); inside changes (\(b,\ h\)) affect \(x\).