Combining Transformations

Pre-Calculus · Axiom Academy

LESSON Combining Transformations When a stretch, a flip, and two shifts all hit one function at once — the order you apply them is everything. 1. Inside the Parentheses vs. Outside Read the master form from the function outward. Whatever sits inside acts on the input — it moves the graph horizontally , and it does the opposite of what it looks like. Whatever sits outside f acts on the output — it moves the graph vertically , exactly as written. Inside (b, h) → reshapes the x-axis Outside (a, k) → reshapes the y-axis 2. The Order That Builds the Graph Watch one parabola, f(x) = x^2 , become . The moves play in the inside → outside order — horizontal first, vertical last — and each frame is a genuine intermediate graph, not a fade. Stretch/compress by , then shift by h . Here b=1 , h=3 : slide right 3. Multiply the height by |a| . Here |a| = 2 : every point gets twice as tall. Here : flip across the x -axis. The parabola now opens downward. Add k last. Here k = 1 : lift the whole graph up 1. You don't have to re-plot the curve to graph a transformation — just push the key points through the same order. Take the point (2, 4) on f(x) = x^2 and feed it through . x-coordinate (inside): b = 1 leaves it alone, then +h adds 3 → the new x is 2 + 3 = 5 . y-coordinate (outside): scale by |a| = 2 gives 8 , reflect ( ) gives -8 , then +k adds 1 → the new y is -7 . sits on f(x)=x^2 , since 2^2 = 4 . sits on g . Check: g(5) = -2(5-3)^2 + 1 = -7 .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.