Step By Step Calculus » 4.6 - Symmetry and Transformations of Functions

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Synopsis
Even Functions: A function is even about aa if f(a-x)=f(a+x)f(a-x)=f(a+x). In particular, a function is simply said to be even if it is even about 00:  f(x)=f(-x)f(x)=f(-x).
Odd Functions: A function is odd about aa if f(a-x) = -f(a+x)f(a-x) = -f(a+x). In particular, a function is simply said to be odd if it is odd about 00:  f(-x) = -f(x)f(-x) = -f(x).
Periodic Functions: A function f(x)f(x) is periodic with period P>0P>0 if f(x + P) = f(x)f(x + P) = f(x) for all values of x\in D_fx\in D_f such that x+P\in D_f.x+P\in D_f.
Composition of Even and Odd Functions: Let \mathbf{o}(x) \mathbf{o}(x) and \mathbf{e}(x) \mathbf{e}(x) stand respectively for any odd or even functions. Then, the following table summarizes the symmetry properties of the four possible compositions:
\begin{tabular}{c|c} even & odd \\\hline $\mathbf{e}\circ \mathbf{o}, \; \mathbf{o}\circ \mathbf{e}, \; \mathbf{e_{1}}\circ \mathbf{e_{2}}$ & $\mathbf{o_1}\circ \mathbf{o_2}$ \end{tabular}
Transformations: The transformations of a function by means of shifting, stretching and reflecting its graph can be defined using composite functions. We know that for any c>0c>0, the graph \{ (x, f(x)) \; | \; x \in D_{f} \}\{ (x, f(x)) \; | \; x \in D_{f} \} of a given function ff is:
  • shifted cc units up or cc units down by replacing f(x) f(x) by f(x)+cf(x)+c  or f(x) -cf(x) -c, respectively. This is the same as saying that the graph of the composite function h(x)=g \circ f (x) = g(f(x))h(x)=g \circ f (x) = g(f(x)), with g(y)=y+cg(y)=y+c or g(y)=y-cg(y)=y-c respectively, is the graph of ff shifted accordingly.
  • shifted cc units left or cc units right by replacing xx by x +cx +c or x - cx - c respectively. This is the same as saying that the graph of the composite function h(x)=f(g(x))h(x)=f(g(x)), with g(x)=x+cg(x)=x+c or g(x)=x-cg(x)=x-c respectively, is the graph of ff shifted accordingly.
  • with c > 1c > 1, stretched or compressed vertically by a factor of cc if y = f(x)y = f(x) is multiplied by cc or \dfrac{1}{c}\dfrac{1}{c} respectively. This is the same as saying that the graph of the composite function
    h(x)=g(f(x))h(x)=g(f(x)), with g(y)=cyg(y)=cy or \displaystyle g(y)=\dfrac{y}{c}\displaystyle g(y)=\dfrac{y}{c} respectively, is the graph of ff stretched or compressed accordingly.
  • with c>1c>1, stretched or compressed horizontally by a factor of cc if xx is replaced by \frac{x}{c}\frac{x}{c} or cxcx respectively. This is the same as saying that the graph of the composite function h(x)=f(g(x))h(x)=f(g(x)) with \displaystyle g(x)=\frac{x}{c}\displaystyle g(x)=\frac{x}{c} or g(x)=cxg(x)=cx respectively, is the graph of ff stretched or compressed accordingly.
  • reflected about the xx-axis if we replace f(x) f(x) by -f(x) -f(x). This is the same as saying that the graph of the composite function h(x)=g(f(x))h(x)=g(f(x)) where g(y)=-yg(y)=-y, is the graph of ff reflected in the xx-axis.
  • reflected about the yy-axis if we replace xx by -x-x. This is the same as saying that the graph of the function h(x)=f(g(x))h(x)=f(g(x)), with g(x)=-x,g(x)=-x, is the graph of ff reflected in the yy-axis.