6.5 Transformation of graphs  (Page 4/5)

Multiplication and division of independent variable

Let us consider an example of functions f(x) and f(2x). The integral values of independent variable are same as integral values on x-axis of coordinate system. Note that independent variable is plotted along x-axis as real number line. The integral 2x values to the function f(2x) - such that input values are same as that of f(x) - are shown on a separate line just below x-axis. The corresponding values are linked with arrow signs. Input to the function f(2x) which is same as that of f(x) now appears closer to origin by a factor of 2. It means graph of f(2x) is same as graph of f(x), which has been shrunk by a factor 2 towards origin. Else, we can say that x-axis has been stretched by a factor 2.

$$y=f\left(x\right)\Rightarrow y=f\left(bx\right);\left|b\right|>1$$

Let us consider another example of functions f(x) and f(x/2). The integral values of independent variable are same as integral values on x-axis of coordinate system. Note that independent variable is plotted along x-axis as real number line. The integral x/2 values to the function f(x/2) - such that input values are same as that of f(x) - are shown on a separate line just below x-axis. The corresponding values are linked with arrow signs. Input to the function f(x/2) which is same as that of f(x) now appears away from origin by a factor of 2. It means graph of f(x/2) is same as graph of f(x), which has been stretched by a factor 2 away from origin. Else, we can say that x-axis has been shrunk by a factor 2.

$$y=f\left(x\right)\Rightarrow y=f\left(\frac{x}{b}\right);\left|b\right|>1$$

Important thing to note about horizontal scaling (shrinking or stretching) is that it takes place with respect to origin of the coordinate system and along x-axis – not about any other point and not along y-axis. What it means that behavior of graph at x=0 remains unchanged. In equivalent term, we can say that y-intercept of graph remains same and is not affected by scaling resulting from multiplication or division of the independent variable.

Negation of independent variable

Let us consider an example of functions f(x) and f(-x). The integral values of independent variable are same as integral values on x-axis of coordinate system. Note that independent variable is plotted along x-axis as real number line. The integral -x values to the function f(-x) - such that input values are same as that of f(x) - are shown on a separate line just below x-axis. The corresponding values are linked with arrow signs. Input to the function f(-x) which is same as that of f(x) now appears to be flipped across y-axis. It means graph of f(-x) is same as graph of f(x), which is mirror image in y-axis i.e. across y-axis.