Function notation & transformations
The idea
Function notation packages a rule under a name — f(x) = x² says feed in x, get back x² — and transformations describe how to move or reshape the entire graph at once. The rigid motions you used on shapes now act on graphs: g(x) = f(x) + 5 shifts up 5, g(x) = f(x − 3) shifts right 3, g(x) = −f(x) reflects across the x-axis, and g(x) = 2f(x) stretches vertically by factor 2. One parent graph plus a short list of moves generates a whole family of functions.
Keep this model: changes outside the function act on outputs and behave as expected, while changes inside act on inputs and run backwards. That inside reversal is the classic misconception — f(x + 3) shifts LEFT, not right, because the input x = −3 now produces what x = 0 used to produce. Order matters as well: in −2f(x + 3) + 5, the stretch and reflection must act before the upward shift. Tracking one landmark point through the moves, in order, keeps a multi-step transformation honest.
Worked example
Let f(x) = x² and g(x) = −2(x + 3)² + 5. List the transformations that turn the graph of f into the graph of g, and find where the point (2, 4) on f ends up.
- Read g as −2 × f(x + 3) + 5. Inside: x + 3 shifts the graph 3 units left. Outside, in order: multiply outputs by 2 (vertical stretch), negate them (reflection across the x-axis), then add 5 (shift up).
- Push the point (2, 4) through: the left shift moves the input from 2 to −1, and the output chain runs 4 → 8 → −8 → −3 after the stretch, reflection, and upward shift.
- Verify with the formula: g(−1) = −2(−1 + 3)² + 5 = −2 × 4 + 5 = −3, so the image (−1, −3) is correct.
- Track the vertex as a second landmark: (0, 0) on f moves to (−3, 5), since stretching and reflecting an output of 0 leaves 0 before the +5. A downward-opening parabola with vertex (−3, 5) and maximum value 5 — the whole picture is consistent.
Answer. g shifts f left 3, stretches it vertically by 2, reflects it across the x-axis, and raises it 5; the point (2, 4) lands at (−1, −3).
Check your understanding
- Why does replacing x with x − 3 move a graph to the right when subtraction feels like it should pull things back?
- How do the graphs of f(2x) and 2f(x) differ, and which points does each transformation leave fixed?
- Why does swapping the order of a vertical stretch and a vertical shift change the result, while two shifts commute?
- How could you reconstruct the full list of transformations just from where the vertex and one other point land?
Build the foundations first
Function notation & transformations builds on these concepts. If any feel shaky, start there.