Calculus I, Section 1.3 | Source: Outlier Guided Notes, Professor Fry
Tags: transforming functions, function transformations, vertical shift, horizontal shift, vertical stretch, vertical shrink, horizontal stretch, horizontal compress, reflection, composition of functions, sin function modelling, graph transformations, calculus 1
Difficulty: Introductory
Prerequisites: Familiarity with basic function notation, the shape of y = sin(x), and how to read a graph.
Function transformations are one of the core tools in calculus for building useful models out of simple base functions. The idea is straightforward: take a function you already understand (like sin(x)) and shift, stretch, compress, reflect or compose it until it fits real data.
This section uses blood pressure modelling as a running example, which makes the abstract rules concrete. If you can transform a function confidently, you will read and write models far more quickly for the rest of the course.
You should already be comfortable with basic function notation (f(x), plugging in values) and recognise what y = sin(x) looks like before working through this.
You can move a graph up/down or left/right by adding constants in the right place, stretch or compress it by multiplying by constants, reflect it by negating, and nest one function inside another to combine behaviours. Every transformation maps to a specific algebraic change in the equation, and learning which change does what is the whole game.
Vertical shift
Adding or subtracting a constant to the entire output of a function: y = f(x) + c. In simple terms, this slides the whole graph up or down without changing its shape.
Vertical stretch / shrink
Multiplying the output of a function by a constant: y = c·f(x). Think of it as pulling the graph taller (c > 1) or squashing it flatter (0 < c < 1) in the y-direction.
Horizontal elongation / squashing (horizontal stretch and compress)
Multiplying the input of a function by a constant: y = f(c·x). In simple terms, this widens the graph (0 < c < 1) or narrows it (c > 1). The effect is the opposite of what most people first expect.
Horizontal shift (phase shift)
Adding or subtracting a constant inside the function's input: y = f(x + c) or y = f(x - c). Think of it as sliding the graph left or right. Adding inside the brackets moves the graph left (counterintuitive), and subtracting moves it right.
Reflection
Negating either the input or the output of a function to mirror it across an axis. y = -f(x) flips the graph over the x-axis; y = f(-x) flips it over the y-axis.
Composition of functions (function of a function)
Replacing the input x with an entire other function, such as x + 0.1x². In simple terms, this lets you layer two behaviours together, for example combining a steady increase with a cyclic pattern.
The base sin(x) function oscillates between -1 and 1. If you are modelling something that is always positive (blood pressure, for instance), you need to lift the entire curve above zero.
y = f(x) + c (c > 0) shifts the graph up by c units
y = f(x) - c (c > 0) shifts the graph down by c units
The shape, width, and height of the curve stay the same; only its vertical position changes
Example: y = sin(x) + 13 centres the wave around 13 instead of 0
The raw sin function only reaches 1 unit above and below its centre. Real-world quantities usually vary by more than that, so you multiply the function's output to widen its range.
y = c·f(x) for c > 1 stretches the graph vertically (taller peaks, deeper troughs)
y = c·f(x) for 0 < c < 1 shrinks the graph vertically (flatter peaks, shallower troughs)
The multiplier scales the amplitude but leaves the horizontal spacing unchanged
Example: y = 3·sin(x) oscillates between -3 and +3. Combined with a vertical shift, y = 3·sin(x) + 13 oscillates between 10 and 16, a reasonable blood pressure range in kPa
When the thing you are modelling speeds up or slows down (a heart rate during sleep, for example), the period of the wave changes. You control this by multiplying the input.
y = f(c·x) for 0 < c < 1 elongates (stretches) the graph horizontally, making the period longer
y = f(c·x) for c > 1 squashes (compresses) the graph horizontally, making the period shorter
This feels backwards at first. The intuition: if c = 0.5, each "new" x-unit covers only half of the original cycle, so the full cycle takes twice as long to complete
Example: y = 3·sin(0.5x) + 13 has a period twice as long as y = 3·sin(x) + 13, modelling the slower heartbeat during sleep
Sometimes the starting value of the model at x = 0 does not match the real data. You fix this by shifting the curve along the x-axis.
y = f(x + c) for c > 0 moves the graph to the left by c units
y = f(x - c) for c > 0 moves the graph to the right by c units
This direction is the opposite of what most students initially guess. The sign inside the brackets works against you: "+" goes left, "-" goes right
Example: in y = 3·sin(0.5x + 1) + 13, the "+1" inside the argument shifts the entire curve one unit to the left
Sometimes a single transformation is not enough. If blood pressure is cyclic but the heart rate is also accelerating (during exercise, say), you can compose functions by replacing x with an entirely different expression.
Start with a function that describes the changing rate: for example, f(x) = x + 0.1x²
Substitute this into the blood pressure model wherever x appears: y = 3·sin(0.5·(x + 0.1x²) + 1) + 13
The outer function (the sin-based model) still handles the oscillation; the inner function (x + 0.1x²) makes the cycles speed up over time
The amplitude of the blood pressure stays the same (still between 10 and 16 kPa), but the peaks come closer together as x increases, reflecting a rising heart rate
A reflection flips the graph across one of the axes.
y = -f(x) reflects the graph over the x-axis (every y-value is negated; peaks become troughs and vice versa)
y = f(-x) reflects the graph over the y-axis (the graph is mirrored left to right)
You can combine reflections with any of the other transformations
Visual shorthand: negating the output flips vertically; negating the input flips horizontally
Transformation | Formula | Effect |
|---|---|---|
Shift up | y = f(x) + c, c > 0 | Graph moves up c units |
Shift down | y = f(x) - c, c > 0 | Graph moves down c units |
Vertical stretch | y = c·f(x), c > 1 | Graph is taller |
Vertical shrink | y = c·f(x), 0 < c < 1 | Graph is flatter |
Horizontal elongate | y = f(c·x), 0 < c < 1 | Graph is wider |
Horizontal squash | y = f(c·x), c > 1 | Graph is narrower |
Shift left | y = f(x + c), c > 0 | Graph moves left c units |
Shift right | y = f(x - c), c > 0 | Graph moves right c units |
Reflect over x-axis | y = -f(x) | Flip vertically |
Reflect over y-axis | y = f(-x) | Flip horizontally |
Running example (final model): y = 3·sin(0.5x + 1) + 13
Breaking this down piece by piece:
+13 shifts the curve up so it centres around 13 kPa
The multiplier 3 stretches vertically, giving a range of 10 to 16 kPa
0.5 inside the argument elongates horizontally, doubling the period
+1 inside the argument shifts the curve one unit to the left
Blood pressure modelling is the example used here, but the same toolkit appears everywhere. Audio engineers shift and stretch waveforms to tune sound. Economists transform base growth curves to fit GDP data. Biologists compose functions when one variable (like temperature) changes another rate (like enzyme activity) that itself drives a cyclic process.
Any time you see a real dataset that looks like a known curve but shifted, scaled or flipped, function transformations are the standard way to fit it.
Students often assume that y = f(x + c) shifts the graph to the right for positive c. It does the opposite: positive c shifts the graph left. The sign inside the brackets works against your intuition.
Horizontal stretch and compress feel backwards. Multiplying x by a number greater than 1 compresses the graph (makes it narrower), because each unit of x now covers more of the original cycle.
Confusing which constant goes where: the vertical transformations (shift and stretch) act on the output of f, while the horizontal transformations act on the input. If you keep this distinction clear, the formulas follow.
Thinking that a vertical stretch also changes the horizontal period. It does not. Vertical and horizontal transformations are independent of each other.
⚠️ Expect questions that give you a transformed equation and ask you to describe each transformation in words. Practice reading y = a·f(b·x + c) + d and naming what a, b, c, and d each do.
⚠️ The direction of horizontal shifts (left vs. right) is one of the most common exam mistakes. Drill this until the counterintuitive sign is automatic.
⚠️ Composition of functions may appear as a modelling question: "given a rate that changes over time, write the combined model." Know how to substitute one function into another.
⚠️ Reflections are often tested alongside symmetry: if f is even, f(-x) = f(x), so the y-axis reflection looks the same.
True or False: y = f(x) + 5 shifts the graph 5 units to the right. (False, it shifts it up.)
Fill in the blank: y = f(x - 3) shifts the graph ______ by 3 units. (Right.)
True or False: Multiplying x by 2 inside f(2x) makes the graph wider. (False, it makes it narrower/squashed.)
Fill in the blank: y = -f(x) reflects the graph over the ______. (x-axis.)
True or False: In y = 3·sin(x) + 13, the 3 changes the period of the wave. (False, it changes the amplitude.)
Q: Given y = sin(x), describe the transformations applied to obtain y = 4·sin(x - 2) + 7.
A: The graph of sin(x) is shifted 2 units to the right (the -2 inside the argument), stretched vertically by a factor of 4 (the multiplier in front), and shifted up 7 units (the +7 added to the output).
Q: Write the equation for sin(x) shifted 5 units up and compressed horizontally by a factor of 3.
A: y = sin(3x) + 5. The 3 multiplies the input (horizontal compression), and the +5 adds to the output (vertical shift up).
Q: Why does y = f(x + c) shift the graph left rather than right when c is positive?
A: Because the function reaches each output value sooner. The point that used to occur at x = a now occurs at x = a - c, which is c units to the left. The "+c" inside the brackets effectively makes x "too big too early," pulling the curve leftward.
Q: In the blood pressure model y = 3·sin(0.5x + 1) + 13, what happens if you change 0.5 to 0.25?
A: The horizontal elongation doubles again, so the period becomes four times the original sin(x) period. Physically, this would represent an even slower heartbeat.
Q: How does composing f(x) = x + 0.1x² into the blood pressure model affect the graph?
A: The cycles of the sin wave get closer together as x increases, because x + 0.1x² grows faster than x alone. The amplitude stays the same (still between 10 and 16 kPa), but the frequency increases over time, modelling a rising heart rate during exercise.
This material connects directly to limits and continuity: when you shift or stretch a function, its limits and points of discontinuity move accordingly, so you need to track transformations when evaluating limits.
It also lays the groundwork for derivatives of composite functions (the chain rule), which is the calculus version of "function of a function." If you are comfortable substituting one function into another here, the chain rule will feel like a natural extension.
In later chapters on integration, recognising a transformed function lets you reverse the transformation and integrate a simpler base function instead.
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