Transformations of Functions
In short: Transformations of functions are changes to a formula, such as shifting, stretching or flipping, that move or reshape the graph of a parent function in predictable ways.
Algebra 2Functions and GraphsF-BF.B.3
What is Transformations of Functions?
The transformations of functions are a small toolkit for turning one known graph into many others. Add a number outside the function and the graph slides up or down; add one inside and it slides sideways, opposite to what the sign suggests.
For a parent function \(f\), the graph of \(g(x) = a\,f(x – h) + k\) is moved right \(h\) units and up \(k\) units, stretched vertically by the factor \(|a|\), and reflected across the x-axis when \(a\) is negative. A factor inside the argument, as in \(f(bx)\), compresses the graph horizontally.
How to spot it
- Transformations of functions are in play when a new formula is built from a familiar function, such as |x| or x squared, with numbers added or multiplied.
- The problem asks how the graph of g compares with the graph of f.
- A table of values is given for f and you must produce a table for a related function.
- A minus sign appears either in front of the function or in front of x inside the parentheses.
The method
- Rewrite the new function in the pattern \(a\,f(x – h) + k\), factoring inside if needed.
- Read \(h\) and \(k\): the graph moves \(h\) units horizontally and \(k\) units vertically.
- Read \(a\): stretch vertically by \(|a|\), and flip across the x-axis if \(a < 0\).
- Move each key point \((x, y)\) of the parent to \((x + h,\ ay + k)\) and connect them.
Transformations of Functions: solved example
Problem
Describe how \(g(x) = -2|x + 3| + 1\) is obtained from \(f(x) = |x|\), and find the vertex of \(g\).
Solution
Write \(g(x) = -2\,f(x – (-3)) + 1\), so \(h = -3\), \(k = 1\) and \(a = -2\).
The graph shifts 3 units left and 1 unit up, is stretched vertically by 2 and is reflected across the x-axis. The vertex \((0, 0)\) moves to \((-3, 1)\), and the graph opens downward.
Try it
Question 1
How does the graph of \(y = (x – 4)^2 – 5\) differ from the graph of \(y = x^2\)?
Show the answer
It moves 4 units right and 5 units down, so the vertex is at \((4, -5)\).
Question 2
A point \((6, 10)\) lies on \(f\). Where is its image on \(y = \frac{1}{2}f(x) + 7\)?
Show the answer
The x-value stays 6 (no inside change) and the y-value becomes \(\frac{1}{2}(10) + 7 = 12\), so the image is \((6, 12)\).
Transformations of Functions vs. Composition of Functions
A transformation changes the picture of a single function by adjusting its formula with shifts, stretches or flips. Composition feeds the output of one function into a second, different function, as in \(f(g(x))\). Replacing \(x\) by \(x – 3\) is a shift, while replacing \(x\) by \(g(x)\) is a composition.
Keep going
Lessons
Worksheets and quizzes
- Transformations of Functions – Worksheet
- Transformations of Functions – Quiz
- Transformations of Graphs – Worksheet
- Transformations of Graphs – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs