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Jim Lamb ers Math 1B Fall Quarter 2004- Le ture 13 Notes
These notes orresp ond to Se tion 5.7 in the text.
Transformations of Fun tions
Mathemati al mo dels typi ally in lude relationships b etween quantities su h as p osition, velo ity, or time, and fun tions, su h as exp onential, logarithmi or trigonometri fun tions, are often ideal for des ribing su h relationships. Su h fun tions that des rib e how a single quantity, alled the dependent variable, dep ends on the value of a se ond quantity, alled the independent variable. In most ases, we will use the letter x to denote an indep endent variable, or t if the indep endent variable is time. The letter y will often b e used to denote the orresp onding dep endent variable. If f (x) is the fun tion that des rib es how y and x are related, then we say that y = f (x); that is, the fun tion f , evaluated at a value x of the indep endent variable, yields the orresp onding value y for the dep endent variable. In the previous le tures, we learned ab out the six basi trigonometri fun tions and their graphs. From these basi fun tions, many other fun tions an b e obtained by applying simple transforma- tions, whi h we now dis uss.
Shifting: adding a onstant to either the indep endent or the dep endent variable auses the graph of a fun tion to shift. Adding a p ositive onstant to the indep endent variable shifts the graph to the left, while adding a negative onstant shifts the graph to the right. Adding a p ositive onstant to the dep endent variable shifts the graph upward, while adding a negative onstant shifts it downward.
Example 1 Figure 1 illustrates horizontal and verti al shifts of the fun tion f (x) = sin x. 2
S aling: Multiplying the indep endent or dep endent variable by a onstant has the e e t of s aling the graph in some manner. For instan e, multiplying the indep endent variable by a onstant , where > 1, ontra ts the graph horizontally by a fa tor of , whereas if 0 < < 1, the graph is stret hed horizontally by a fa tor of 1 =. Similarly, if the dep endent variable is multiplied by a onstant , where > 1, the graph is stret hed verti ally by a fa tor of , whereas it ontra ts verti ally by a fa tor of 1 = if 0 < < 1.
Example 2 Figure 2 illustrates horizontal and verti al s aling of the fun tion f (x) = sin x. 2
−1 0 1 2 3 4 5 6 7
−0.
0
1
x
y
Horizontal shifts of y=sin(x) sin(x) sin(x+π/4) sin(x−π/2)
−3 0 1 2 3 4 5 6 7
−
−
0
1
2
Vertical shifts of y=sin(x)
x
y
sin(x) sin(x)+ sin(x)−
Figure 1: Top plot: horizontal shifts of sin x. Note that adding a p ositive onstant to x shifts the graph to the right, while using a negative onstant shifts to the left. Bottom plot: verti al shifts of sin x. Note that adding a p ositive onstant to the dep endent variable shifts the graph up, while a negative onstant shifts the graph down.
Graphing y = k + A sin(B x + C ) and y = k + A os(B x + C )
Many appli ation areas deal with waves, su h as sound waves, radio waves, o ean waves, or light waves. Su h waves are des rib ed using fun tions in whi h the dep endent variable ontinually os illates within a small range of values, in a y li al pattern, as the indep endent variable hanges. Typi ally, su h fun tions have the form f (x) = k + A sin(B x + C ) or f (x) = k + A os (B x + C ), where A, B , C , and k are onstants, and B > 0. Fun tions of this form are known as simple harmoni s, and they are used to des rib e simple harmoni motion. We already know from the previous le ture that if y = sin x or y = os x, then y os illates inde nitely b etween 1 and 1 as x hanges, and this os illation o urs in a y le that rep eats itself every 2 units, sin e the sine and osine fun tions are b oth 2 -p erio di. However, what if we wanted to use these fun tions to des rib e a wave that os illated in a di erent pattern? For
Next, we shift the graph horizontally to the left by C =B units if C =B > 0, or to the right by C =B units if C =B < 0. The numb er C =B , whi h represents the numb er of units that the graph is shifted to the right, is alled the phase shift of the urve. Next, we s ale the graph verti ally. If jAj > 1, then we stret h the graph verti ally by a fa tor of jAj, whereas if 0 < jAj < 1, then we ontra t the graph verti ally by a fa tor of 1 =jAj. In addition, if A < 0, the graph is also re e ted ab out the x-axis. The numb er jAj is alled the amplitude of the urve. The y -values of p oints on the urve deviate from the average y -value by jAj in either dire tion. Finally, we shift the graph verti ally. If k > 0, then we shift the graph upward by k units, whereas if k < 0, then we shift the graph downward by jk j units.
Example 3 We will onstru t the graph of the simple harmoni
y = 4 3 os (2x 6) (2)
from the graph of the \simpler harmoni " y = os x using graphi al transformations. First, we rewrite the equation (2) in the form
y = 4 3 os [2(x 3)℄ (3)
so that the phase shift an easily b e seen to b e 3. Then, we p erform the following transformations:
- rst, we ontra t the graph horizontally by a fa tor of 2, to obtain the graph of y = os (2x).
- we shift the graph of y = os (2x) to the right by 3 units, to obtain the graph of y = os [2(x 3)℄.
- we stret h the graph verti ally by a fa tor of 3, to obtain the graph of y = 3 os [2(x 3)℄.
- we re e t the graph ab out the x-axis, to obtain the graph of y = 3 os [2(x 3)℄.
- nally, we shift the graph upward by 4 units, to obtain the graph of y = 4 3 os [2(x 3)℄.
These transformations are shown in Figure 3. 2
−10 −5 0 5 10
−
0
5
y=cos x
−10 −5 0 5 10
−
0
5
y=cos(2x)
−10 −5 0 5 10
−
0
5
y=cos(2*(x−3))
−10 −5 0 5 10
−
0
5
y=3cos(2(x−3))
−10 −5 0 5 10
−
0
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y=−3cos(2(x−3))
−10 −5 0 5 10
−
0
5
y=4−3cos(2(x−3))
Figure 3: Transformation of y = os x into y = 4 3 os [2(x 3)℄, in stages. Upp er left: original graph y = os x. Upp er right: ontra ting horizontally by fa tor of 2 yields y = os (2x). Middle left: shifting by 3 units to the right yields y = os [2(x 3)℄. Middle right: stret hing verti ally by fa tor of 3 yields y = 3 os [2(x 3)℄. Lower left: re e ting ab out x-axis yields y = 3 os [2(x 3)℄. Lower right: shifting upward by 4 units yields nal graph of y = 4 3 os [2(x 3)℄.
