How to Transform Linear Functions: Rules and Examples
Transform a linear function by changing its inputs, outputs, or both. For \(f(x)=mx+b\), adding outside the function shifts it vertically; changing the input shifts or scales it horizontally. Write the transformed rule before simplifying so the operation stays clear. Rotation needs a specified center and can produce a vertical line, which is not a function of x.
Transform Linear Functions: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Identify the operationWrite the transformation using f before expanding. State the reflection axis or rotation center when needed.
- Substitute the original ruleReplace f with its complete expression and keep parentheses around changed inputs or outputs.
- Check a pointMap a point from the original line and verify that it satisfies the transformed equation.
Worked examples
Shift right 3 units
Start with \(f(x)=2x+1\).
- Write \(g(x)=f(x-3)\).
- Substitute: \(g(x)=2(x-3)+1\).
- Simplify: \(g(x)=2x-5\).
The original point \((0,1)\) moves to \((3,1)\). Check: \(g(3)=1\).
Reflect, then shift up 4
Start with the same function.
- Reflect across the x-axis: \(-f(x)=-2x-1\).
- Add 4: \(g(x)=-f(x)+4\).
- Simplify: \(g(x)=-2x+3\).
The point \((0,1)\) becomes \((0,-1)\), then \((0,3)\). Order matters: \(-(f(x)+4)\) is a different rule.
Try one before moving on
Transform Linear Functions: pop-up practice
Related Topics
- How to Transform Quadratic Equations
- Transformation on the Coordinate Plane: Dilation
- Transformation on the Coordinate Plane: Rotation
- Transformation on the Coordinate Plane: Reflection
- Transformation: Rotations, Reflections, and Translations
Step-by-step to Transform Linear Functions
These rules all begin with \(f(x)=2x+1\). Keep the original function visible while comparing them.
| Operation | New rule | Point check |
|---|---|---|
| Up 2 | \(f(x)+2=2x+3\) | \((0,1)\to(0,3)\) |
| Right 3 | \(f(x-3)=2x-5\) | \((0,1)\to(3,1)\) |
| Reflect across x-axis | \(-f(x)=-2x-1\) | \((0,1)\to(0,-1)\) |
| Reflect across y-axis | \(f(-x)=-2x+1\) | \((1,3)\to(-1,3)\) |
| Horizontal stretch by 2 | \(f(x/2)=x+1\) | \((1,3)\to(2,3)\) |
| Halve horizontal distances | \(f(2x)=4x+1\) | \((1,3)\to(1/2,3)\) |
| Vertical stretch by 2 | \(2f(x)=4x+2\) | \((1,3)\to(1,6)\) |
Why the rotation center matters
A 90-degree counterclockwise rotation about the origin maps \((x,y)\) to \((-y,x)\). Rotating \(y=2x+1\) this way gives \(y=-\tfrac12x-\tfrac12\): the original point \((0,1)\) becomes \((-1,0)\). The equation \(y=-\tfrac12x+1\) instead describes a 90-degree rotation about the original y-intercept, \((0,1)\).
A negative reciprocal slope identifies a perpendicular direction when the original slope is nonzero; it does not specify a rotation center or intercept. A horizontal line rotated 90 degrees becomes vertical and fails the vertical-line test. Ordinary input/output shifts, nonzero scalings, and axis reflections preserve a nonvertical straight-line graph; arbitrary rotations require this extra check.
Reference: OpenStax: transformations of functions. The rotation equations above are verified by mapping points directly.
Related to This Article
More math articles
- CLEP College Algebra Math Practice Test Questions
- 3rd Grade NYSE Math FREE Sample Practice Questions
- Grade 6 ELA Tests: A Clear Parent Guide to Reading, Writing, and Test-Day Readiness
- 8th Grade SBAC Math Worksheets: FREE & Printable
- Global Inequality and Development
- Free Georgia Grade 2 English Worksheets
- Grade 3 Math: Multiplication Facts (0-5)
- Lobbying
- Number Navigators: How to Select Pairs with Targeted Sums and Differences
- The Best Grade 4 Math Book for Arkansas Students
What people say about "How to Transform Linear Functions"?
No one replied yet.