Concept 6

Apply transformations (shifts, reflections, stretches/compressions) to graphs and equations.

Transformations shift, reflect, stretch, or compress a graph or equation.

Core Idea

Outside changes affect 𝑦-values; inside changes affect 𝑥-values. Vertical moves read naturally, but horizontal moves run in the opposite direction from the sign inside the function.

Understanding

Rule: For transformations, separate the rule into parts before you interpret it. In 𝑔(𝑥) = −2𝑓(𝑥 +3) +5, the +5 moves the graph up, the −2 reflects it across the 𝑥-axis and stretches it vertically by a factor of 2, and the 𝑥 +3 shifts the graph left 3 units.

Students often mix up horizontal direction because they read the sign inside too quickly. A helpful check is to ask what input to 𝑓 gives the old point. If the inside is 𝑥 +3, then the same old output now appears 3 units earlier on the 𝑥-axis, so the graph moves left.

Step by Step

  1. Read outside constants and shifts first: they change vertical position and vertical size.
  2. Read the expression inside the parentheses second: it controls horizontal movement or horizontal scaling.
  3. Watch for a negative sign outside the function for reflection across the 𝑥-axis.
  4. For horizontal shifts, reverse the direction you might expect from the sign inside.
Question

Worked Example

The graph of 𝑦 =𝑓(𝑥) is transformed to 𝑦 = −2𝑓(𝑥 +3) +5. Which description is correct?

Select an answer to see the explanation