# What effect does multiplication by a constant have on a function's plot, in general?

I am looking at the equation of a line and its plot. I can intuitively see what these operations do to the plot's form: adding a constant, multiplying by a constant, adding a parametrised value, etc. For some simple conic functions, I am able to do the same, almost as intuitively, although not always. When I consider higher order polynomials, the effects vary to the point of not being able to recognize the original plot anymore. Is there a general definition for what effect multiplying for a constant has on a function's plot? I am using gnuplot and/or octave to play around with this (should I be using something else?) and still, for some function, I can see the effects and understand them (e.g. This "amplifies the oscillation range", this "contracts" this part of the curve, etc. but is there a more general way of making estimations about what will happen to the function's plot?)

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When you say "multiplication by a constant", do you mean $y=k\times f(x)$, $y= f(k \times x)$, or something else? – Henry Aug 20 '12 at 20:48
y = f(x) k, sorry. Is k not called a multiplicative constant in this case? – Robottinosino Aug 20 '12 at 20:50
If $k > 1$ (resp. $0 < k < 1$) then it stretches (resp. shrinks) the plot in the $y$-axis direction. If $k$ is negative then it's the same $+$ mirroring about $x$-axis. – user2468 Aug 20 '12 at 21:02

For a positive value $k$, the graph of $y = kf(x)$ is obtained from the graph of $y=f(x)$ by scaling in the vertical direction. In coordinates, if $(x_0, y_0)$ is on the graph of $y=f(x)$, then that point moves to $(x_0, ky_0)$ on the graph of $y=kf(x)$. Any point on the $x$-axis remains fixed (as then $y=0$). When $k > 1$, the graph is stretched vertically (away from the $x$-axis), while if $0 < k < 1$, the graph is compressed vertically (toward the $x$-axis).
Another way to think about this: The graph of $y=kf(x)$ is exactly the same as the graph of $y=f(x)$ if the $y$-axis is scaled so that $1$ is relabeled $k$, $2 \mapsto 2k$, $-1 \mapsto -k$, etc.