Graphing y = f(x-3) + 2

Pre-Calculus · Axiom Academy

EXAMPLE Graphing y = f(x − 3) + 2 Reading a horizontal and a vertical shift straight off the equation The parent function is f(x) = x^2 . Graph the transformed function g(x) = (x - 3)^2 + 2, describe the shift inside and outside the function, and find where the vertex lands. Same parabola shape, new location — shifted right 3 and up 2. The vertex slides from (0, 0) to (3, 2). Nice work. You read both shifts straight off g(x) = (x - 3)^2 + 2 and tracked the vertex to its new home. Inside the function: (x - h) shifts the graph right by h — opposite to the sign you see. Here x - 3 means right 3. Outside the function: +k shifts the graph up by k . Here +2 means up 2. Vertex form: g(x) = (x - h)^2 + k has its vertex at (h, k) , so this one sits at (3, 2) with axis of symmetry x = 3 . Shape is unchanged: every point moves the same way — right 3, up 2 — so the parabola keeps its width and only its position changes. Once you can spot h and k in the equation, you can graph any shifted parabola without plotting a single point.

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