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MathidiaMathematics Academy
Advanced graphs and transformations
Lesson 21 of 75
Introduction

Transformations of f(x)

Objective. Translate, reflect and stretch graphs using algebraic notation.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Transformations of f(x) describe how changing a function's rule shifts, reflects or stretches its graph: f(x)+a moves the graph up by a, f(x+a) moves it left by a, −f(x) reflects it in the x-axis, f(−x) reflects it in the y-axis, and af(x) stretches it vertically by scale factor a. Each transformation acts on the output or input of the function in a specific, predictable way that can be read directly from the modified equation. Combining transformations requires applying them in the correct order, usually matching the order operations would be carried out on x. For example, the graph of y=f(x−2)+3 is the graph of y=f(x) translated 2 units right and 3 units up.

Worked example

The graph of y=f(x) has a minimum point at (2, −1). State the coordinates of the minimum point on the graph of y=f(x−4)+5.

2 … 1
Coordinate grid with the line 2 … 1
  1. 1Identify the transformation: f(x−4)+5 translates the graph 4 units right and 5 units up
  2. 2Apply the horizontal shift to the x-coordinate: 2+4=6
  3. 3Apply the vertical shift to the y-coordinate: −1+5=4
  4. 4Combine both shifted coordinates

Answer: (6, 4)

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Common mistake. Students often transpose the direction of horizontal translations, believing f(x+3) shifts the graph right by 3 when it actually shifts left by 3; remember that changes inside the brackets act in the opposite direction to their sign.

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