Key takeaway: Transformations of functions are changes to the equation of a parent function that shift, reflect, stretch or compress its graph in a predictable way.
Grade 11 – Algebra 2F-BF.B.3Functions and Graphs
Definition of Transformations of Functions
Transformations of functions take a known graph, the parent function, and move or reshape it by editing its formula. Changes made outside the function, such as adding a number to the output, act on the vertical direction. Changes made inside the parentheses act on the horizontal direction, and they often behave opposite to what a beginner expects.
One equation can combine several moves. A reliable order is horizontal shift first, then reflection and stretch, and finally the vertical shift. Learning transformations of functions lets you sketch a new graph from a few key points of the old one, with no table of values.
Formula and notation
With a parent function \(f\), the general form is
\[g(x)=a\,f(x-h)+k.\]
Here \(h\) shifts right (left if negative), \(k\) shifts up, \(|a|\) stretches vertically when it exceeds 1, and a negative \(a\) reflects over the x-axis. Replacing \(x\) with \(-x\) reflects over the y-axis.
Key facts
- The point (h, k) is where the vertex or key corner of the parent graph lands.
- Inside changes work backward: f(x + 5) moves the graph left, not right.
- A factor with 0 < |a| < 1 compresses the graph toward the x-axis.
- Shifts and reflections keep the shape; stretches and compressions change it.
Transformations of Functions example
Describe how \(g(x)=-3(x+5)^2+8\) is built from the parent \(f(x)=x^2\), and find \(g(-4)\).
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Write \(x+5\) as \(x-(-5)\): shift 5 units left.
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The factor \(-3\) reflects over the x-axis and stretches vertically by 3.
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The \(+8\) shifts the graph 8 units up, so the vertex is \((-5,8)\).
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Evaluate: \(g(-4)=-3(1)^2+8=5\).
Left 5, reflect, stretch by 3, up 8; \(g(-4)=5\).
Common mistake
Watch out
Elena claims that \(y=(x+5)^2\) is the parent parabola moved 5 units to the right.
Why it is wrong
The vertex is where the inside equals zero: \(x+5=0\) gives \(x=-5\). The graph moves 5 units to the left. The inside operation is the reverse of what the sign suggests.
Related terms and lessons
- Algebra 2 · Transformations of FunctionsLessonWorksheetQuiz
- Algebra 1 · Transformations, Absolute Value and Piecewise FunctionsLessonWorksheetQuiz
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