Graphical Transformations for IPY3Y4 Math

> **Q:** What does Graphical Transformations for IPY3Y4 Math cover?\
> **A:** Step by step guide on graphical transformations

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:

1. Move the graph up or down by (a) units.
2. Move the graph right or left by (a) units.
3. Multiply all (y)-coordinates by (a) (negative (a) also flips over the (x)-axis).
4. Multiply all (x)-coordinates by (1/a) (negative (a) also flips over the (y)-axis).
5. Multiplying all (y)-coordinates by (-1) flips the graph over the (x)-axis.
6. 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](https://eclatinstitute.sg/blog/ip-maths-tuition) 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](https://www.stitz-zeager.com/szprecalculus07042013.pdf)