Transformations of Graphs
This topic covers how the graph of a function y = f(x) is altered by simple, predictable changes to its equation. Understanding these rules allows for rapid sketching and analysis of related functions without needing to plot individual points, a key skill for non-calculator exams.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Transformations outside the function bracket, like f(x) + a and a·f(x), affect the graph vertically (affecting y-coordinates).
- Transformations inside the function bracket, like f(x + a) and f(ax), affect the graph horizontally (affecting x-coordinates) and often work in a counter-intuitive way.
- y = f(x) + a is a vertical translation. The graph shifts up if a > 0 and down if a < 0.
- y = a·f(x) is a vertical stretch by a factor of 'a'. If a < 0, it also reflects the graph in the x-axis.
- y = f(x + a) is a horizontal translation. The graph shifts LEFT if a > 0 and RIGHT if a < 0.
- y = f(ax) is a horizontal stretch by a factor of '1/a'. If a < 0, it also reflects the graph in the y-axis.
- The order of combined transformations is critical. Typically, stretches and reflections should be applied before translations.
Diagram
Formulae
y = f(x) + a To translate the graph of f(x) vertically by 'a' units. Shift by vector (0, a).
y = f(x + a) To translate the graph of f(x) horizontally by '-a' units. Shift by vector (-a, 0).
y = a f(x) To stretch the graph of f(x) vertically by a scale factor of 'a', measured from the x-axis.
y = f(ax) To stretch the graph of f(x) horizontally by a scale factor of '1/a', measured from the y-axis.
Definitions
- Translation
- A transformation that shifts every point on a graph by the same distance in the same direction, without changing its shape or orientation.
- Stretch
- A transformation that scales a graph away from a fixed line (an axis) by a constant factor. This changes the graph's shape.
- Reflection
- A transformation that 'flips' a graph across a line of reflection, such as the x-axis or y-axis. This is equivalent to a stretch with a scale factor of -1.
- Function Composition (f(g(x)))
- The process of applying one function to the result of another. The output of the inner function, g(x), becomes the input for the outer function, f.
Worked example
The graph of y = f(x) passes through the point P(4, -6). What are the coordinates of the corresponding point P' on the graph of y = -2f(x - 3)?
- 1
Identify the transformations by comparing y = -2f(x - 3) to the base function y = f(x).
- 2
The '-3' inside the bracket corresponds to f(x + a) with a = -3.
This is a horizontal translation of -(-3) = +3 units (3 units to the right).
- 3
The '-2' outside the bracket corresponds to a·f(⋯) with a = -2.
This is a vertical stretch by a factor of 2, and a reflection in the x-axis.
- 4
Apply the horizontal transformation to the x-coordinate of P(4, -6):
x' = 4 + 3 = 7 - 5
Apply the vertical transformations to the y-coordinate of P(4, -6):
y' = -6 × (-2) = 12 - 6
Combine the new coordinates to find the point P'.
Answer: The new coordinates are P'(7, 12).
Common mistakes
- ×Sign error on horizontal shifts: For y = f(x + 5), the shift is 5 units to the LEFT, not the right. Remember that the transformation inside the bracket is counter-intuitive.
- ×Reciprocal error on horizontal stretches: For y = f(2x), the graph is compressed horizontally by a factor of 1/2, not stretched by a factor of 2. You need to halve your x-values to get the same output.
- ×Incorrect order for combined transformations: For y = f(ax + b), which can be written f(a(x + b/a)), the horizontal stretch by 1/a happens before the horizontal shift by -b/a. Applying them in the wrong order gives an incorrect result.
No-calculator tips
- ✓To check your work, consider what input makes the argument of the new function equal to a key input of the old function. For y=f(x-3), the original behavior at x=0 now occurs when x-3=0, i.e., at x=3. This confirms a shift right by 3.
- ✓Track key points individually. Instead of trying to visualize the whole curve transforming, just move the vertices, intercepts, and asymptotes one transformation at a time.
- ✓Think about the effect on coordinates. For y' = a f(x') + c, the new y-coordinate is simply y' = a*y + c. For y' = f(bx' + d), you must solve for x: x = bx' + d, so x' = (x - d)/b.