Transformations of Graphs – Shifting, Stretching and Reflecting
Once you know the shape of a graph like y = x² or y = 2x, you rarely need to plot every new related graph from scratch. A small change to the equation – adding a number, changing a sign, multiplying by a constant – moves, flips, or stretches the original shape in a completely predictable way. Learning these transformation rules turns graphing from a slow plotting exercise into something you can often do in your head.
This systematic approach to transforming graphs grew directly out of Descartes's coordinate geometry: because every equation is tied to a specific set of points, changing the equation in a structured way changes the points – and therefore the picture – in an equally structured way. Function notation (f(x)), introduced by Euler, is what makes it possible to describe these rules in general terms that apply to any function, not just one specific equation.
Vertical Translations
y = f(x) + k shifts the graph up by k units (or down, if k is negative). Adding to the outside of the function changes every output (y-value) by the same amount.
Horizontal Translations
y = f(x − h) shifts the graph right by h units. This one catches people out because the sign appears to flip: changing the input requires x to be h larger to produce the same output as before, which moves the whole graph to the right.
Compared to y = x², this graph is translated 3 units right and 2 units up. The vertex moves from (0, 0) to (3, 2).
Reflections
y = −f(x) reflects the graph in the x-axis (flips it upside down).
y = f(−x) reflects the graph in the y-axis (flips it left to right).
Replacing x with −x reflects the graph in the y-axis, turning growth into decay – y = 2−x is the mirror image, decreasing from left to right.
Stretches
y = a · f(x) stretches the graph vertically by a factor of a (if a > 1) or compresses it (if 0 < a < 1). A negative a combines a stretch with a reflection in the x-axis.
Quick Reference
| Change to f(x) | Effect on the Graph |
|---|---|
| f(x) + k | Shift up by k |
| f(x) − k | Shift down by k |
| f(x − h) | Shift right by h |
| f(x + h) | Shift left by h |
| −f(x) | Reflect in the x-axis |
| f(−x) | Reflect in the y-axis |
| a · f(x), a > 1 | Stretch vertically by factor a |
Real-Life Application
Transformations aren't just abstract algebra: a sound engineer adjusting volume is applying a vertical stretch to a waveform graph; a physicist modelling a delayed reaction is applying a horizontal shift to a graph of concentration over time; and a designer mirroring a logo is applying a reflection.
Key Takeaways
- f(x) + k and f(x) − k shift a graph vertically up or down.
- f(x − h) and f(x + h) shift a graph horizontally right or left – the sign appears to flip.
- −f(x) reflects in the x-axis; f(−x) reflects in the y-axis.
- a · f(x) stretches the graph vertically by a factor of a.