Graph transformations – translations, stretches and reflections of y = a·f(b(x − h)) + k
MathematicsFunctions & GraphsAges 15–16
Loading…
Sign in to playPick a parent function (x, x², x³, |x|, √x, 1/x, 1/x², 2ˣ, eˣ, ln x, sin x) and drag a, b, h and k in g(x) = a·f(b(x − h)) + k to see stretches, reflections and translations. Arrows join key points to their images, and Show steps replays the transformations in the correct order. The Related graphs screen draws |g(x)|, g(|x|), 1/g(x) and [g(x)]², and the Symmetry screen tests for even, odd and self-inverse functions.
Lesson: Transformations of graphs; even and odd functions
What it shows
A parent function f is transformed into g(x) = a·f(b(x − h)) + k. Each point (x, y) of f maps to (x/b + h, a·y + k): a horizontal stretch with scale factor 1/b (with a reflection in the y-axis if b < 0), a horizontal translation by h, a vertical stretch with scale factor a (with a reflection in the x-axis if a < 0) and a vertical translation by k. Domain, range, asymptotes, amplitude and period follow from a, b, h and k. Even, odd and self-inverse properties are checked numerically for x from −6 to 6, not proved algebraically.
How to use
On Transform, choose a Parent function and drag the a, b, h and k sliders, or drag on the graph to translate it. Tick Key points to see where each point goes, and press Show steps to replay the transformations in order. On Related graphs, choose a Graph such as y = 1/g(x) and drag x₀. On Symmetry, choose a Test and compare the dashed image with g. Reset restores the starting values.
Parameters you can change
- Screen Transform, Related graphs, Symmetry
- Parent function f f(x) = x, f(x) = x², f(x) = x³, f(x) = |x|, f(x) = √x, f(x) = 1/x, f(x) = 1/x², f(x) = 2ˣ, f(x) = eˣ, f(x) = ln x, f(x) = sin x
- Coefficient a (vertical stretch; a < 0 reflects in the x-axis) -4–4
- Coefficient b (horizontal stretch, scale factor 1/b; b < 0 reflects in the y-axis; b ≠ 0) -3–3
- Horizontal translation h -5–5
- Vertical translation k -5–5
- Show key points and arrows to their images
- Related graph (Related graphs screen) y = |g(x)|, y = g(|x|), y = 1/g(x), y = [g(x)]²
- Property tested (Symmetry screen) Even: g(−x) = g(x), Odd: g(−x) = −g(x), Self-inverse: g(g(x)) = x
- x-coordinate x₀ of the chosen point (Related graphs and Symmetry screens) -5–5
Questions to explore
- Why is the graph of y = (2x − 4)² a translation of y = (2x)² by 2 units, not 4 units?
- Is y = 1/(x − 2) + 2 self-inverse, and in which line is its graph symmetric?
- When you draw y = 1/g(x), what happens to the zeros of g and to the points where g(x) = ±1?
