Functions: Transformations
Lessons in this chapter
About Functions: Transformations
Functions: Transformations applies the transformation ideas from geometry to any function's graph.
Translations of Graph of Functions covers sliding a graph up, down, left, or right by adding or subtracting a constant to the function or its input, while Reflections of Graphs of Functions covers flipping a graph over the x-axis or y-axis by negating the function or its input.
Scaling and Combinations of Transformations closes the chapter by stretching or compressing a graph vertically or horizontally, then combining multiple transformations: translations, reflections, and scaling together, into a single equation.
This is a short chapter, and each of its three lessons is an Algebra 1 worksheet worked in the page: Translations of Graph of Functions, Reflections of Graphs of Functions, and Scaling and Combinations of Transformations. The questions are scored as you answer them, which is the quickest way to find out whether a shift inside the brackets is being read as a horizontal move or a vertical one. Free to try, with no login needed for your first 3.
Frequently asked questions
Translations of Graph of Functions covers shifting a graph up, down, left or right without changing its shape, and Reflections of Graphs of Functions flips it across the x-axis or the y-axis. Scaling and Combinations of Transformations then stretches or compresses the graph and applies several transformations in sequence, where the order they are carried out in changes the result. The chapter sits in the high school Functions conceptual category (F).
Adding a constant outside the function (like f(x) + 3) shifts the entire graph vertically: up for a positive constant, down for a negative one.
Adding a constant inside the function’s input (like f(x + 3)) shifts the graph horizontally, but in the opposite direction of what the sign suggests: f(x + 3) shifts LEFT, not right.
Negating the whole function (-f(x)) reflects the graph over the x-axis; negating the input (f(-x)) reflects it over the y-axis.
Multiplying the function by a constant greater than 1 stretches it vertically; multiplying by a constant between 0 and 1 compresses it, multiplying the INPUT works the opposite way, affecting the graph horizontally.
Assuming a horizontal shift or stretch moves in the same direction the equation suggests, since transformations to the input work backward from what’s intuitive, f(x - 2) shifts RIGHT, not left, which trips up students expecting it to match vertical transformations’ intuitive direction.