Drawing Graph of the Function y=f(kx)

Task 1. Draw graph of the function y=f(kx), where k>0,k!=1, knowing graph of the function y=f(x).

Let y_0=f(x_0). Now answer the following question: what value of argument x should we take, so function y=f(kx) will take value y_0? Clearly this value should satisfy the following condition: kx=x_0 or x=(x_0)/k. Therefore, point (x_0;y_0) that lies on graph of the given function y=f(x) is trasformed into point ((x_0)/k;y_0) that lies on the graph of the function y=f(kx). This transformation is called compressing of graph y=f(x) with coeffcient k to y-axis (if 0<k<1 then, in fact, we stretch from y-axis with coeffcient 1/k).

On figure to the right you can see graphs of the following functions: y=arccos(x) and y=arccos(2x); y=sqrt(x) and y=sqrt(x/3). In first case, we compress graph to y-axis because k=2>1; in second case we stretch graph because 0<k=1/3<1.

Task 2. Draw graph of the function y=f(-x), knowing graph of the function y=f(x).

Let y_0=f(x_0). Function y=f(-x) will take value y_0 if argument x satisfies following condition: x_0=-x, i.e. x=-x_0. Point of the graph (x_0;y_0) of graph of the function y=f(x) is transformed into point (-x_0;y_0) of graph of the function y=f(-x). This means that we can obtain graph of the function y=f(-x) by reflecting graph of the function y=f(x) about y-axis.

On figure to the left you can see graphs of functions y=log_3(x) and y=log_3(-x).

Task 3. Draw graph of the function y=f(kx), where k<0,k!=-1, knowing graph of the function y=f(x).

We have that f(kx)=f(-|k|x). Therefore, to obtain graph of the function y=f(kx) we first compress graph of the function y=f(x) to y-axis with coeffcient |k| and the reflect result about y-axis.

On the figure to the right you can see graphs of function y=x^(3/2) and y=(-2x)^(3/2).