Graphical Transformations for IPY3Y4 Math
Q: What does Graphical Transformations for IPY3Y4 Math cover?
A: Step by step guide on graphical transformations
TL;DR
Graph transformations become easier when you separate outside changes from inside changes.
Outside changes affect y-values directly. Inside changes affect x-values and often feel reversed.
Use a small set of anchor points before drawing the whole graph.
Quick transformation map
- Outside changes move or stretch y: Look at what happens after f(x).
- Inside changes move or scale x: Check the sign carefully because shifts can feel reversed.
- Track three anchor points: Transform the points first, then sketch the curve.
Concrete example: If a graph shifts right by 2 and up by 3, move a known point such as (1, 4) to (3, 7). Do this for two or three points before drawing the final curve.
Status: Refreshed 2025-12-15. Source link checked; examples remain aligned to standard IP Y3–Y4 graph transforms.
Quick links: IP Math hub, IP Math roadmap
Base graph: (y = f(x))
/* prettier-ignore */| Number | Transformation | Effect on graph |
|---|---|---|
| 1 | (y = f(x) + a) | Shift up by (a) units (down if (a < 0)) |
| 2 | (y = f(x + a)) | Shift left by (a) units (right if (a < 0)) |
| 3 | (y = af(x)) | Vertical stretch/compression by (\lvert a\rvert); reflect in the x-axis if (a < 0) |
| 4 | (y = f(ax)) | Horizontal scale by (1 / \lvert a\rvert); reflect in the y-axis if (a < 0) |
| 5 | (y = -f(x)) | Reflection about the x-axis |
| 6 | (y = f(-x)) | Reflection about the y-axis |
In layman terms:
- Move the graph up or down by (a) units.
- Move the graph right or left by (a) units.
- Multiply all (y)-coordinates by (a) (negative (a) also flips over the (x)-axis).
- Multiply all (x)-coordinates by (1/a) (negative (a) also flips over the (y)-axis).
- Multiplying all (y)-coordinates by (-1) flips the graph over the (x)-axis.
- Multiplying all (x)-coordinates by (-1) flips the graph over the (y)-axis.
Rule of thumb: transformations on (x) act "inside" the function, so additions feel reversed (add (a) (\rightarrow) move left (a)) and multiplications invert ((\times a) (\rightarrow) scale by (1/a)). (y)-axis moves follow the sign you see.
Multiple transformations (work inside the brackets first, then outside):
/* prettier-ignore */| Number | Transformation | Effects on graph |
|---|---|---|
| 1 | (y = c f(bx + a) + d) | "AMMA": shift left by (a), scale (x) by (1 / \lvert b\rvert) (reflect if (b < 0)), then scale (y) by (c) and shift up by (d). |
Need more structured walkthroughs and practice sets? Continue with our IP Maths hub for WA calendars, topic plans, and links to the full transformation drills.
Sources:
- Stitz, Carl J., and Zeager, Jeff. Precalculus (Revised 7 Apr 2013), Section 1.7 on graph shifts, reflections, and scalings (pp. 120–131): https://www.stitz-zeager.com/szprecalculus07042013.pdf
