Transformations of parent functions
Transformations of Parent Functions
Transformations of Parent Functions is a topic in Relations & Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.2(A), which requires students to graph the parent functions and analyze the effect of parameter changes on their graphs.
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 \(x\).