Transformations of Functions
Shifts, stretches and reflections change a parent graph.
About this topic
Transformations move and reshape a graph without changing its basic form. In y = a·f(b(x − h)) + k, h shifts left or right, k shifts up or down, a stretches or compresses vertically (negative a flips over the x-axis), and b stretches horizontally (negative b flips over the y-axis).
Changes inside the brackets act on x and seem backwards: f(x − 3) moves right. Knowing these rules lets you sketch complicated graphs from a few parent functions like x², abs(x) and √x.
Worked example: start with the parent y = x². The function y = −2(x − 3)² + 5 moves the vertex to (3, 5), stretches the parabola vertically by a factor of 2, and flips it upside down so it opens downward. Apply the changes in a sensible order: horizontal shift, stretch or flip, then vertical shift. A vertex at (3, 5) with a downward opening is easy to sketch with just a few extra points.
Animators and game designers use the same ideas to move, scale and mirror shapes on screen. The most common slip is the direction of horizontal shifts: f(x + 2) moves the graph left, not right, because x must be 2 smaller to give the same output. Also remember that a factor b inside compresses horizontally by 1/b, which feels opposite to the vertical stretch. Test one point to check your sketch.
Key ideas
- f(x) + k shifts up
- f(x − h) shifts right
- −f(x) flips over the x-axis
Quick check
Use cases: where Transformations of Functions helps
- Homework: review the key idea behind Transformations of Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Transformations of Functions without notes.
- Visual learning: play with the interactive lab to see how Transformations of Functions changes when you adjust its inputs.
- Teaching a friend: explain Transformations of Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Transformations of Functions?
Shifts, stretches and reflections change a parent graph.
What will I learn in Transformations of Functions?
f(x) + k shifts up. f(x − h) shifts right. −f(x) flips over the x-axis.
Is Transformations of Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Transformations of Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Transformations of Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.