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Operations on graphs

Part 2 · Functions · Chapter 9 · lecture notes by Fabio Furini · Chapter PDF

1. Operations on graphs

  • Knowing the graph of a function \(y = f (x)\), by means of simple geometric transformations it is possible to draw the graph of the following functions:
\[\begin{align} y_1 &= f(x) + a, \quad a \in \mathbb{R}\\[2ex] y_2 &= f(x + a), \quad a \in \mathbb{R}\\[2ex] y_3 &= k \: f(x), \quad k \in \mathbb{R}\\[2ex] y_4 &= f(k \:x), \quad k \in \mathbb{R}\\[2ex] y_5 &= |f(x)| \\[2ex] y_6 &= f(|x|) \end{align}\]
  • Therefore, starting from the elementary functions, it is possible to build, by means of these operations, a large variety of new functions. These operations create translations, dilations and reflections of the original function.

Example 1: Operations on graphs related to \(y_1= f(x) +a\)

Consider \(y = \ln x\). Then

\[ y_1 = \ln (x) +a ~~~~(x > 0). \]

Figure 1

  • The graph of \(y_1\) is obtained from that of \(y\) by a translation of \(a\) units upward if \(a> 0\), downward if \(a < 0\).

Example 2: Operations on graphs related to \(y_2= f(x +a)\)

Consider again \(y = \ln x\). Then

\[ y_2 = \ln (x +a). \]

Since \(\ln x\) is defined for \(x > 0\), \(\ln(x +a)\) will be defined for \(x +a > 0\), i.e., for \(x > - a\). Since \(\ln x =0\) if \(x = 1\), we have \(\ln(x +a) =0\) for \(x +a= 1\), i.e., \(x = 1 - a\).

Figure 2

  • The graph of \(y_2\) is obtained from that of \(y\) by a translation of \(a\) units to the left if \(a > 0\), to the right if \(a < 0\).

  • The graph of \(y_3 = k \: f ( x)\) is obtained from that of \(f\) by multiplying all the ordinates \(f(x)\) by \(k\).

  • In particular, if \(k = -1\) the ordinates simply change sign, so that the graph of \(y_3\) is symmetric to that of \(f\) with respect to the \(x\)-axis.

  • Note that if \(k > 1\), the graph is “stretched” in the vertical direction, dilating the positive ordinates upward and the negative ones downward.

  • Conversely, if \(0 < k < 1\) the graph “shrinks”, again in the vertical direction.

  • Therefore, the operation of multiplying \(f ( x)\) by \(k\) has the geometric meaning of a dilation (if \(|k| > 1\)) or a contraction (if \(|k| < 1\)) along the \(y\)-axis, possibly accompanied by a reflection with respect to the \(x\)-axis, if \(k < 0\).

Example 3: Operations on graphs related to \(y_3= k\: f( x)\)

Consider \(y = \sin x\), then \(y_3 = k\: \sin(x).\)

Figure 3

Figure 4

Figure 5

  • The graph of \(y_4 = f ( k\:x)\) is obtained from that of \(f(x)\) by a change of scale along the \(x\)-axis.

  • If \(k > 1\), \(k \:x\) grows faster than \(x\), and therefore the graph of \(y_4\) will be similar to that of \(f\) but with faster oscillations, that is, it will be “compressed” in the horizontal direction by a factor \(\frac{1}{k}\).

  • Similarly, if \(0 < k < 1\) the graph will appear “dilated” in the horizontal direction, with gentler oscillations.

  • If \(k < 0\), in addition to a compression (if \(|k| > 1\)) or a dilation (if \(|k|< 1\)) along the \(x\)-axis, there will be a reflection with respect to the \(y\)-axis.

Example 4: Operations on graphs related to \(y_4 = f(k\: x)\)

Consider \(y = \sin x\), then \(y_4 = \sin(k\: x).\) With \(k=\frac{1}{2}\) and \(k=2\) we have:

Figure 6

Figure 7

Example 5: Operations on graphs related to \(y_4= f(k\: x)\)

Consider \(y = \ln x\), then \(y_4 = \ln(k\: x).\) With \(k=-1\) we have:

Figure 8

Absolute value of \(f(x)\):

\[ |f(x)|= \begin{cases} f(x) & {\rm if~~} f(x)\ge0,\\ -f(x) \: & {\rm if~~} f(x)<0 \end{cases} \]

  • When passing from the graph of \(y=f(x)\) to that of \(y_5=|f(x)|\), the points with non-negative ordinate remain unchanged, while those with negative ordinate are transformed into their symmetric points with respect to the \(x\)-axis.

  • The graph of \(y_5=|f(x)|\) is obtained from that of \(f\) by “flipping” symmetrically with respect to the \(x\)-axis the part of the graph of \(f\) that lies in the lower half-plane, and leaving the rest unchanged.

Example 6: Operations on graphs related to \(y_5= |f(x)|\)

Consider \(y = x\), then \(y_5 = |x|\)

Figure 9

Figure 10

Example 7: Operations on graphs related to \(y_5= |f( x)|\)

Consider \(y = \sin x\), then \(y_5 = |\sin x|\)

Figure 11

Figure 12

  • Finally, to draw the graph of \(y_6 =f(|x|)\), we observe that \(|x| = x\) for \(x \ge 0\), and hence in the right half-plane the two graphs coincide.

  • We have \(|-x|=|x|\), and hence \(y_6\) is an even function, therefore symmetric with respect to the \(y\)-axis.

  • Consequently, the graph of \(y_6 = f(|x|)\) is drawn by leaving the graph of \(f\) unchanged in the right half-plane and flipping it symmetrically with respect to the \(y\)-axis.

Example 8: Operations on graphs related to \(y_6=f(|x|)\)

Consider \(y = e^x\), then \(y_6 = e^{|x|}\)

Figure 13

Figure 14

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