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Integrated Math II
15. CC.HSF.BF.B.3
15.2 Transformations of functions: Horizontal translations
Horizontal Translations of Functions
See exactly how replacing x with x minus h or x plus h slides a graph left or right, with rules, graphs, and worked examples.
What You'll Learn
A horizontal translation slides a graph left or right without changing its shape or size.
Replacing x with x minus h in f(x) shifts the graph right by h units when h is positive.
Replacing x with x plus h shifts the graph left by h units, which is the opposite direction of what the sign suggests.
Horizontal translations change the domain of a function but never its range.
Combining a horizontal shift with a vertical shift moves a graph diagonally while keeping its shape identical.
What You'll Practice
1
Shifting quadratic functions left and right by modifying x-values
2
Sketching translated graphs using the three-point method with intercepts
3
Comparing original and translated functions to identify direction and distance of shift
4
Working with expressions like f(x - a) and f(x + a) to transform graphs
Why This Matters
Horizontal translations are essential for understanding how functions transform across coordinate planes. You'll use this skill constantly in algebra, trigonometry, and calculus when working with shifted parabolas, sine waves, and other function families. Mastering this concept helps you quickly visualize and graph complex functions.
Before You Start — Make Sure You Can:
This Unit Includes
2 Video lessons
Learning resources
Skills
Horizontal Translation
Function Transformations
Graph Shifting
Coordinate Plane
Quadratic Functions
Function Notation
OH Curriculum Aligned