Graph Transformations: Shifts, Reflections, Stretching
Graph transformations change the position or shape of a function graph in predictable ways: shifts move it, reflections flip it and stretching or compressing changes its steepness. In brief: f(x) + k shifts the graph up by k, f(x − h) shifts it right by h, −f(x) reflects it in the x-axis and a · f(x) stretches it vertically by the factor |a|. For example, y = (x − 3)2 + 1 is the parabola y = x2 moved 3 right and 1 up.
Instead of making a table for every new function, you can start with a known parent graph and transform it. This is fast and shows the structure of the formula. This guide covers each transformation, how to combine them in the right order and gives a summary table and a worked example.
How do vertical and horizontal shifts work?
Vertical shift: y = f(x) + k moves every point of the graph k units up if k > 0 and |k| units down if k < 0. The point (a, b) becomes (a, b + k). The change is outside the function and it acts on the outputs, so the effect is as expected.
Horizontal shift: y = f(x − h) moves the graph h units to the right if h > 0 and to the left if h < 0. The point (a, b) becomes (a + h, b). The effect seems reversed: f(x + 3) moves the graph 3 units to the left. The reason is that the input x must be 3 smaller to produce the same value as before: x + 3 = a means x = a − 3.
Example: for f(x) = |x|, the graph of |x − 2| + 1 is a V with vertex (2, 1).
How do reflections work?
Reflection in the x-axis: y = −f(x) changes the sign of every output. The point (a, b) becomes (a, −b). A parabola opening up becomes one opening down.
Reflection in the y-axis: y = f(−x) changes the sign of every input. The point (a, b) becomes (−a, b). For f(x) = √x, the graph of √(−x) is defined for x ≤ 0 and lies to the left. An even function is unchanged by this reflection.
How do stretching and compressing work?
Vertical stretch: y = a · f(x) multiplies every output by a. For |a| > 1 the graph is stretched away from the x-axis, and for 0 < |a| < 1 it is compressed toward it. The point (p, q) becomes (p, aq). Zeros do not change.
Horizontal stretch: y = f(bx) divides every x-coordinate by b. For |b| > 1 the graph is compressed horizontally, and for 0 < |b| < 1 it is stretched. The point (p, q) becomes (p/b, q). Again the effect is the reverse of what the sign may suggest: f(2x) is half as wide.
Graph transformations summary table
| New function | Transformation | Point (x, y) becomes |
|---|---|---|
| f(x) + k | Shift up by k | (x, y + k) |
| f(x) − k | Shift down by k | (x, y − k) |
| f(x − h) | Shift right by h | (x + h, y) |
| f(x + h) | Shift left by h | (x − h, y) |
| −f(x) | Reflect in the x-axis | (x, −y) |
| f(−x) | Reflect in the y-axis | (−x, y) |
| a · f(x), a > 0 | Vertical stretch or compression | (x, ay) |
| f(bx), b > 0 | Horizontal stretch or compression | (x/b, y) |
In which order do you apply several transformations?
For y = a · f(b(x − h)) + k, a reliable order is: first the horizontal changes (stretch by b, then the shift by h), then the vertical ones (stretch by a, then the shift by k). The shifts come last because they should not be scaled.
Worked example: y = −2√(x + 1) + 3
- Start with the parent y = √x, which has key points (0, 0), (1, 1), (4, 2).
- Shift 1 left: (−1, 0), (0, 1), (3, 2).
- Stretch vertically by 2: (−1, 0), (0, 2), (3, 4).
- Reflect in the x-axis: (−1, 0), (0, −2), (3, −4).
- Shift up by 3: (−1, 3), (0, 1), (3, −1).
The graph starts at (−1, 3) and falls to the right through (0, 1) and (3, −1). The domain is x ≥ −1 and the range is y ≤ 3. Check at x = 3: −2√4 + 3 = −1 ✓.
Second example: y = (x − 3)2 + 1
From y = x2, shift 3 right and 1 up. The vertex moves from (0, 0) to (3, 1), and the point (1, 1) of the parent becomes (4, 2).
Where do these ideas come from?
The study of how an equation changes when the variables are shifted goes back to the introduction of analytic geometry in the seventeenth century, when Descartes and Fermat showed that curves can be moved by changing their equations. The systematic use of a parent function and a transformation language became widespread in school mathematics in the twentieth century, supported by graphing technology.
Where are transformations used today?
Computer graphics move, mirror and scale shapes with the same ideas, written as matrices. In science, a graph of a model is fitted to data by adjusting shift and stretch parameters, such as the amplitude and phase of a wave. Sound editing, image scaling and animation all use shifts, reflections and stretches.
Which mistakes are common?
Shifting the wrong way horizontally
f(x + 3) moves the graph left. Think of the input needing to be 3 smaller.
Wrong order of operations
In −2√(x + 1) + 3, shifting up before stretching gives a different graph. Apply the multiplication first and the addition last.
Applying a horizontal stretch to the wrong factor
f(2x) compresses the graph to half the width, it does not stretch it. Divide the x-coordinates by 2.
Forgetting that reflection changes the range
After −f(x), the old maximum becomes a minimum and the range flips.