Vertical stretch is the transformation f(x) = a g(x) that multiplies every y-value by a constant. In Calculus I, it changes a graph’s height or amplitude without changing its domain.
Vertical stretch is a function transformation in Calculus I that multiplies every output by the same constant. If you start with y = g(x) and write y = a g(x), the new graph is scaled vertically by the factor a.
When a is greater than 1, the graph stretches away from the x-axis. Every point gets taller, so peaks rise higher and valleys fall lower. When 0 < a < 1, the graph compresses toward the x-axis, which makes the graph flatter. If a is negative, you still get a vertical scaling, but the graph also reflects across the x-axis.
The biggest thing to keep straight is that vertical stretch changes y-values, not x-values. That means the domain stays the same, because the input values are untouched. What changes is the range, along with features like maximum height, minimum height, and amplitude for wave-like graphs.
A quick example makes the pattern easy to see. If f(x) = x^2 and you stretch it by 3, then the new function is y = 3x^2. The point (2, 4) becomes (2, 12), and the point (-1, 1) becomes (-1, 3). The graph keeps the same basic shape, but it is narrower because the outputs grow faster.
For trig graphs, vertical stretch is often described as amplitude. If y = sin x has amplitude 1, then y = 4 sin x has amplitude 4. That means the wave reaches from -4 to 4 instead of from -1 to 1. In Calculus I, this comes up when you compare function families, sketch graphs, or track how transformations change the look of a curve before you ever differentiate it.
Vertical stretch shows up everywhere Calculus I asks you to read a function from its graph or predict a graph from its equation. If you can spot the multiplier in front of a function, you can tell whether the graph gets taller, shorter, or reflected, which saves time on graphing questions.
It also connects to the basic classes of functions you meet early in the course. A stretched quadratic still looks like a parabola, a stretched exponential still keeps exponential growth behavior, and a stretched trig graph still keeps the same period while its amplitude changes. That means you can separate shape from scale.
This matters in curve sketching too. If you know a graph has been stretched vertically, you can focus on how the outputs change instead of restarting from scratch. That makes it easier to compare related functions, interpret transformations in a family of graphs, and check whether an equation matches a picture.
A lot of common mistakes come from mixing up vertical stretch with horizontal compression. In Calculus I, those are different moves with different effects, so recognizing which variable gets multiplied is a big deal. Vertical stretch is one of the first places where the algebra of a function turns directly into its geometry.
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view galleryTransformation
Vertical stretch is one kind of transformation. It changes the graph of a parent function without changing the basic function family, so you can still recognize the original shape after the scaling. In Calculus I, transformations are how you build and compare new graphs from familiar ones.
Amplitude
For trigonometric functions, vertical stretch changes amplitude. If you multiply sine or cosine by a constant, the wave gets taller or shorter, but its period stays the same. That is why amplitude is the easiest way to describe vertical stretch on oscillating graphs.
Dilation
A vertical stretch is a dilation in the y-direction. The whole graph is scaled by the same factor, so every output changes proportionally. This idea shows up when you compare coordinate pairs before and after a transformation.
leading coefficient
For polynomial functions, the leading coefficient can act like a vertical stretch when you compare graphs with the same degree. A larger absolute value usually makes the graph steeper or narrower, while a smaller absolute value makes it flatter. That is one reason coefficient size matters in graph shape.
A problem set or quiz question will usually give you a function like y = 2f(x) or y = 1/3 x^2 and ask what happens to the graph. Your job is to multiply the y-values, describe whether the graph is stretched or compressed, and say whether the domain changes. If the factor is negative, mention the reflection too.
You may also be asked to identify amplitude on a sine or cosine graph, or to match an equation to a sketch. In those cases, look for the multiplier outside the function, not the exponent or the x-value. That is the move that tells you whether the graph got taller or shorter.
These are easy to mix up because both change the look of a graph. Vertical stretch multiplies the outputs, so it changes y-values and amplitude. Horizontal compression changes the inputs, so it squeezes the graph left and right instead. If you see the constant outside the function, think vertical. If you see it attached to x inside the function, think horizontal.
A vertical stretch multiplies every output of a function by the same constant.
If the factor is greater than 1, the graph gets taller, and if it is between 0 and 1, the graph gets shorter.
The domain stays the same because only the y-values change.
For sine and cosine graphs, vertical stretch changes amplitude.
The fastest way to identify it is to look for a constant outside the function, like a g(x) or a f(x).
Vertical stretch is a transformation that multiplies a function’s outputs by a constant. In Calculus I, it changes the graph’s height or amplitude without changing the x-values. You will usually see it written as y = a f(x).
No, vertical stretch does not change the domain because the input values stay the same. It only changes the output values. What does change is the range, and sometimes the amplitude or overall steepness of the graph.
Vertical stretch changes y-values, while horizontal compression changes x-values. A vertical stretch uses a constant outside the function, and a horizontal compression uses a constant inside the function with x. That difference tells you whether the graph is getting taller or narrower sideways.
For sine and cosine graphs, amplitude is the absolute value of the coefficient in front of the function. So if y = 3 sin x, the amplitude is 3. If the coefficient is 1/2, the graph is compressed and the amplitude is 1/2.