In AP Precalculus, a multiplicative transformation multiplies a function's output (g(x) = af(x)) or input (g(x) = f(bx)) by a nonzero constant, producing a vertical or horizontal dilation of the graph, plus a reflection over the x-axis or y-axis when the constant is negative (CED 1.12.A).
A multiplicative transformation changes a function by multiplication instead of addition. There are two flavors, and the difference is where the constant sits. If you multiply the output, you get g(x) = af(x), which vertically dilates the graph of f by a factor of |a|. Every y-value gets stretched (if |a| > 1) or compressed (if |a| < 1). If a is negative, the graph also flips over the x-axis. If you multiply the input, you get g(x) = f(bx), which horizontally dilates the graph by a factor of 1/|b|. That reciprocal trips people up, but it makes sense once you see it. A factor like b = 2 means the function reaches each output twice as fast, so the graph squeezes to half its width. If b is negative, the graph reflects over the y-axis.
Here's the intuitive split worth memorizing. Multiplication stretches and flips; addition slides. Multiply the output and the graph changes vertically. Multiply the input and it changes horizontally, by the reciprocal. The CED also gives you a deeper way to see this in Topic 2.7. A multiplicative transformation is just a composition with the simple function g(x) = kx. Writing f(2x) as f composed with "double it" explains why the input gets processed before f ever sees it, and why the effect on the graph runs backward.
This term lives in Topic 1.12 (Transformations of Functions) under learning objective 1.12.A, which asks you to construct functions that are additive and/or multiplicative transformations of other functions. But it doesn't stay in Unit 1. Topic 2.7 reframes it through composition (2.7.C.3 says a multiplicative transformation can be understood as composing f with g(x) = kx), and Topic 3.8 applies it to the tangent function, where 3.8.C.3 describes g(θ) = a tan θ as a vertical dilation by |a| with an x-axis reflection when a < 0. In Unit 3 more broadly, the constants in a sin(b(x + c)) + d are literally multiplicative transformations wearing trig names. Amplitude is a vertical dilation, and period change is a horizontal dilation. If you can read af(bx + c) + d fluently, every sinusoidal modeling problem gets easier.
Keep studying AP® Precalculus Unit 1
Horizontal dilation (Units 1 and 3)
This is the trickier half of multiplicative transformations. In g(x) = f(bx), the graph dilates by 1/|b|, the reciprocal of what you'd guess. So f(3x) compresses the graph to one-third its width. In Unit 3, this same move is what changes the period of sine, cosine, and tangent.
Function composition and f(g(x)) (Unit 2)
CED 2.7.C.3 says a multiplicative transformation is just f composed with g(x) = kx. Seeing f(2x) as f(g(x)) explains why input changes act 'inside out.' The doubling happens before f runs, so the effect on the graph is the reverse of what the formula seems to say.
The tangent function and its asymptotes (Unit 3)
For g(θ) = a tan θ, the |a| vertically dilates the graph and a negative a reflects it over the x-axis (3.8.C.3). But tangent has no amplitude because its range is all reals, so the dilation shows up as steeper or shallower growth between asymptotes, not a taller wave.
Function decomposition (Unit 2)
Decomposition runs the composition idea in reverse. Given something like h(x) = ln(5x), you should be able to peel it apart as f(x) = ln x composed with g(x) = 5x, recognizing the 5x as a multiplicative transformation hiding inside a more complicated function.
Multiplicative transformations show up most often in multiple choice, where you identify what type of transformation an equation represents or match a transformed equation to its graph. A classic stem looks like this. Given h(x) = -2f(x), name the transformation. The answer combines two pieces, a vertical dilation by a factor of 2 and a reflection over the x-axis because the constant is negative. You also need the skill in reverse, constructing the equation when given a verbal description like 'horizontally compressed by a factor of 1/3.' In Unit 3, the same idea gets tested through trig. Identifying amplitude and period from a sin(bx) is a multiplicative transformation question in disguise. No released FRQ uses the phrase verbatim, but the modeling FRQs constantly require you to build functions of the form af(b(x + c)) + d, which means correctly placing dilation constants is a free-response skill, not just an MCQ one.
Additive transformations add a constant (f(x) + k or f(x + h)) and slide the graph without changing its shape. Multiplicative transformations multiply by a constant (af(x) or f(bx)) and stretch, compress, or reflect the graph. Quick test on an MCQ stem. If the constant is added, it's a translation. If it's multiplied, it's a dilation, possibly with a reflection. The exam loves asking you to classify an equation as one or the other, so check the operation before anything else.
A multiplicative transformation multiplies a function or its input by a nonzero constant, which dilates the graph instead of translating it.
Multiplying the output, as in g(x) = af(x), vertically dilates the graph by a factor of |a|, and a negative a adds a reflection over the x-axis.
Multiplying the input, as in g(x) = f(bx), horizontally dilates the graph by the reciprocal factor 1/|b|, and a negative b reflects the graph over the y-axis.
Per CED 2.7.C.3, every multiplicative transformation is secretly a composition of f with the simple function g(x) = kx.
In Unit 3, amplitude and period changes for sinusoidal functions are multiplicative transformations, and for tangent, a tan θ dilates vertically by |a| since tangent has no amplitude.
If the constant is added, the graph slides (additive); if the constant is multiplied, the graph stretches or flips (multiplicative).
It's a transformation where a function or its input is multiplied by a nonzero constant, written g(x) = af(x) or g(x) = f(bx). The result is a vertical or horizontal dilation of the graph, with a reflection if the constant is negative (CED 1.12.A.3).
No, it compresses it. The horizontal dilation factor is the reciprocal, 1/|b|, so f(2x) squeezes the graph to half its original width. The function reaches each output value twice as fast.
Additive transformations add a constant and translate (slide) the graph, like f(x) + k or f(x + h). Multiplicative transformations multiply by a constant and dilate or reflect the graph, like af(x) or f(bx). The operation tells you which one you're looking at.
Yes. Reflections come from negative constants in multiplicative transformations. In g(x) = af(x), a negative a reflects over the x-axis; in g(x) = f(bx), a negative b reflects over the y-axis. So h(x) = -2f(x) is both a vertical dilation by 2 and an x-axis reflection.
They're the same idea with trig vocabulary. In a sin(bx), the |a| is a vertical dilation that sets the amplitude, and the b is a horizontal dilation that sets the period at 2π/|b|. For tangent, 3.8.C.3 says a tan θ vertically dilates by |a|, but tangent has no amplitude because its range is unbounded.
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