Function Transformations

Function transformations let you take a familiar graph and move, flip, or resize it without changing its basic shape family. They are important because they help students recognize patterns instead of memorizing many separate graphs. Once you know a parent function like y = x^2 or y = |x|, you can predict many related graphs quickly. This skill is used in algebra, precalculus, physics, and data modeling.

Most transformations come from changing the equation in specific ways. Adding or subtracting outside the function causes vertical shifts, while changes inside the input cause horizontal effects that work in the opposite direction from what many students expect. Negative signs create reflections, and multiplying by constants creates stretches or compressions. By tracking a few key points and understanding the rules, you can sketch transformed graphs accurately.

Key Facts

Vertical shift: y = f(x) + k moves the graph up k units, and y = f(x) - k moves it down k units.
Horizontal shift: y = f(x - h) moves the graph right h units, and y = f(x + h) moves it left h units.
Reflection across the x-axis: y = -f(x).
Reflection across the y-axis: y = f(-x).
Vertical stretch or compression: y = a f(x). If |a| > 1, stretch vertically. If 0 < |a| < 1, compress vertically.
Horizontal stretch or compression: y = f(bx). If |b| > 1, compress horizontally by factor 1/|b|. If 0 < |b| < 1, stretch horizontally by factor 1/|b|.

Vocabulary

Parent function: The simplest form of a function family, such as y = x^2 for quadratics.
Translation: A transformation that slides a graph left, right, up, or down without changing its shape.
Reflection: A transformation that flips a graph across an axis.
Vertical stretch: A change that makes a graph taller by multiplying all y-values by a factor greater than 1.
Horizontal compression: A change that squeezes a graph toward the y-axis by changing the input with a factor greater than 1.

Common Mistakes to Avoid

Treating y = f(x + 3) as a shift right 3, which is wrong because horizontal changes inside the function work in the opposite direction, so it shifts left 3.
Confusing y = -f(x) with y = f(-x), which is wrong because the first reflects across the x-axis while the second reflects across the y-axis.
Thinking y = 2f(x) and y = f(2x) do the same thing, which is wrong because the first changes y-values vertically and the second changes x-values horizontally.
Applying transformations in the wrong place in the equation, which is wrong because outside changes affect outputs and inside changes affect inputs, leading to different graphs.

Practice Questions

1 Start with the parent function y = x^2. Describe the transformations that produce y = (x - 4)^2 + 3, and state the new vertex.
2 The graph of y = |x| is transformed to y = -0.5|x + 2| - 1. Identify the horizontal shift, vertical shift, reflection, and vertical stretch or compression.
3 Explain why y = f(x - 5) and y = f(x) - 5 are different transformations, and describe how each changes the graph.