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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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.
Hope this helps!