Download section 1.6 families of functions

AP CALCULUS NOTES
SECTION 1.6 FAMILIES OF FUNCTIONS
A.) Families of Lines: A set or family of lines consists of all lines that share one of the
following four characteristics:
 They represent a constant function which means they are all horizontal lines.
 They share the same slope.
 They share the same y-intercept.
 They all pass through a particular point.
Examples:
Horizontal Lines
Lines with slope = 2
Lines with y-int = 4
Lines through  2, 1
y  2 x  b
yc
y  mx  4
y  1  m  x  2
B.) The Family of Power Functions y  x n where n is constant share the following
characteristics:
 For even values of n, the functions f  x   xn are even, and their graphs are symmetric

about the y-axis. The graphs have the same general shape as the parabola y  x 2 (though
they are not actually parabolas if n  2 ).
For odd values of n, the functions f  x   xn are odd, and their graphs are symmetric

about the origin. The graphs have the same general shape as y  x3 .
For all values of n, the graphs pass through the origin and the point 1,1 . For even
values of n, the graphs pass through  1,1 . For odd values of n, the graphs pass through
 1, 1 .

Increasing n causes the graph to become flatter over the interval 1  x  1 and steeper
over the intervals x  1 and x  1.
Example: pg. 76
14.a.)
y  2 x n where n  1,3,5
y  2 x n where n  2, 4,6
How would you explain why this last characteristic is true?
C.) The Family of Power Functions y  x  n 
1
where n is constant share the following
xn
characteristics:
 For even values of n, the functions f  x   x  n are even, and their graphs are symmetric
1
.
x2
are odd, and their graphs are symmetric
about the y-axis. The graphs have the same general shape as the y 

For odd values of n, the functions f  x   x  n
1
.
x
For all values of n, the graphs pass through the point 1,1 and have a break (called a
about the origin. The graphs have the same general shape as the y 

discontinuity) at x  0 . The discontinuity is caused because we cannot divide by 0.
For even values of n, the graphs pass through  1,1 . For odd values of n, the graphs
pass through  1, 1 .

Increasing n causes the graph to become steeper over the intervals 1  x  0 and
0  x  1 and flatter over the intervals x  1 and x  1.
1
1
y  n where n  1,3,5
y  n where n  2, 4, 6
Examples:
x
x
How do these graphs compare with the ones in sample HW problem 14b?
1
D.) The Family of Power Functions y  x n  n x where n is constant share the following
characteristics:
 For odd values of n, the graphs of y  n x have the same general shape as y  3 x . The
graphs extend over the entire x-axis. Note: f  x   3 x is defined for all real values of x.

For even values of n, the graphs of y  n x have the same general shape as y  x . The
graphs only extend over the nonnegative x-axis.
Examples: translated left 2 units, vertically stretched by a factor of 3, reflected over x-axis
14.c.)
y  3  x  2 
1
n
where n  1,3,5
y  3  x  2 
1
n
where n  2, 4, 6
E.) Polynomial Functions - Let n be a nonnegative integer and let an , an1 ,..., a1 , a0 (coefficients)
be real #’s with an  0 . The function given by f  x   an xn  an1 xn1  ...  a2 x2  a1x  a0 is
called a polynomial function of x with degree n.
domain :  ,  
1.) Degree = 0 implies a constant function.
f  x   a, a  0
Ex.) f  x   2
2.) Degree = 1 implies a linear function
f  x   ax  b, a  0
Ex.) f  x   2 x  1
3.) Degree = 2 implies a quadratic function
f  x   ax2  bx  c, a  0 Ex.) f  x   2 x2  x
Ex.) f  x   5x3  2
f  x   ax3  bx2  cx  d
Note: graphs of polynomials have no discontinuities or sharp corners.
p  x
F.) Rational Functions - can be written in the form f  x  
where p  x  and q  x  are
q  x
4.) Degree = 3 implies a cubic function
polynomials.
Domain: all values of x such that q  x   0


Graphs of rational functions have discontinuities at the points where the denominator is 0.
Rational functions may have numbers at which they are not defined. Near such points,
many (but not all) rational functions have graphs that approximate a vertical line called a
vertical asymptote.
 The graphs of many (but not all) rational functions eventually get closer and closer to
some horizontal line called a horizontal asymptote as one travels along the curve in either
the positive or negative direction.
p  x  an x n  an 1 x n 1  ...  a1 x  a0

TO Find ASYMPTOTES: Let f  x  
where p  x  and
q  x  bm x m  bm 1 x m 1  ...  b1 x  b0
q  x  have no common factors. Assume p  x  has degree n and q  x  has degree m.
1.) The graph of f  x  has vertical asymptotes at the zeroes of q  x  .
2.) The graph of f  x  has at most 1 horizontal asymptote as follows:
a.) If n  m , the x-axis  y  0  is a horizontal asymptote.
a
b.) If n  m , the line y  n is a horizontal asymptote.
bm
c.) If n  m , the graph of f  x  has NO horizontal asymptote.
Example: pg.77
x 1
x 1
27.b.) y  2
VA: x  3, x  2

x  x  6  x  3 x  2 
Graph I
HA: y  0 (deg of num < deg of den)
G.) Trigonometric Functions – Let  be a real # (angle measure) and (x,y) be the point on the
unit circle x 2  y 2  1 corresponding to arc length s  r .
y
x
1
1
sin  
csc   , y  0
cos  
sec   , x  0
1
1
x
y
y
, x0
x
x
, y0
y


Examples: cot 0 and csc 0 are undefined. tan
, sec
are undefined.
2
2
Trigonometric Functions are periodic. They behave in a cyclic or repetitive manner. A function
f is periodic if there exists a positive real # c such that f   c   f   for all  in the domain
of f. The least value of c for which f is periodic is the period of f.
This section mainly focuses on properties on sine and cosine:
 As the point P  x, y    cos ,sin   moves around the unit circle, the coordinates vary
tan  

cot  
between –1 and 1   1  sin   1 and  1  cos   1
sine and cosine have a period of 2  sin   2 n   sin and cos   2 n   cos
 See Appendix E at the back of the text for a more thorough review of properties.
Graphs of y  d  a sin  bx  c  or y  d  a cos bx  c 
y  sin x
y  cos x
1. The a  is called the AMPLITUDE.
a: constant scaling factor  vertical stretch if a  1 or vertical shrink if a  1
Range : a  y  a Reflection in x – axis for y  a sin x or y  a cos x
2. b : let b be a positive real #, b affects the period  horizontal stretch if 0  b  1
horizontal shrink if b  1
2
b
period 
frequency 
b
2
3. c : involved in horizontal or phase shift -  bx  c  implies shift c units to the right
b
bx  c  implies shift c b units to the left
4. d : represents vertical shift. Graph oscillates about the horizontal line y  d called the
axis of the wave.
Hints for sketching sine and cosine:
1.) Find endpoints of 1 period:
Left Endpoint: set bx  c  0 and solve for x.
Right Endpoint: set bx  c  2 and solve for x.
2.) To find intervals between 5 key points of one period take the period  4.
Helpful Hints for determining mathematical models of the form:
y  d  a sin  bx  c  or y  d  a cos bx  c 
1.) Amplitude (a): Calculate the amplitude of the wave by taking
max  min
.
2
2.) Period: cosine – take the difference between 2 maximum values and divide that difference by
the number of cycles.
sine – take the difference between 2 points on the axis of the wave and divide that
difference by the number of cycles.
2
b – use the formula period 
.
b
max  min
3.) Vertical Shift (d): Calculate the axis of the wave by taking
.
2
4.) Phase shift: cosine – since this function starts at a maximum value when x = 0, the location
of a maximum determines the actual phase shift.
sine – since this function starts at the axis of the wave when x = 0, the location of a
point on the axis of the wave prior to a max determines the phase shift.
c
c – use the formula phase shift = .
b
Example: pg. 78
max  min
max  min
 1 (axis of the wave) and A 
2
41.c.) min  1, max  3  D 
2
2
2
period   
C  0 (no phase shift for cos)
starts at min   cos x
B
B2
Equation: y  1  2 cos 2 x Since we want an equation in terms of sin, we know

that sin involves a phase shift of
and starts at the axis of the wave. Thus,
2


y  1  2sin  2 x   .
2

Sample Homework Problems: pgs. 75-78
2x  9 y  7  0
8.) 5 x  3 y  11  0
2x  7
5 x  11
y
y
point of intersection:
9
3
1

 2, 
3

1

Using point-slope form, the equation for the family of lines that pass through the point  2,  is
3

1
1
y   m  x  2  or y   m  x  2  .
3
3
14.b.)
y   x  n where n  1,3,5
y   x  n where n  2, 4, 6
10
8
17.b.) y  3  x  2  translated right 2 units, vertically stretched
by a factor of 3, reflected about the x-axis,
original: y  x3
6
3
4
2
-2
-1
1
2
3
4
-2
-4
-6
-8
-10
1
translated right 2 units, translated up 1 unit,
x2
1
original: y 
x
20.a.) y  1 
8
6
4
2
-8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8
-2
-4
-6
-8
24.) rectangular container that has a square base
 Volume  100cm3  x2 y where x  side of base, y  height of container
100
y
x2
Area = 2 (area of the base) + 4 (area of a side)
400
 100 
 2  x 2   4  xy   2 x 2  4 x  2   2 x 2 
x
 x 
100
 4.6416cm
Calculating the min of the graph, we get x  4.6416cm . y 
2
 4.6416 
Thus, a cube with sides 4.6416 cm would require the least amount of material.
3
 270 coordinates on a unit circle that represent  :  0, 1
30.a.)  
2
sin   1, cos  0, tan   undef , csc  1, sec  undef , cot   0
34.a.) 56  20  36
36
1

This implies the ship travels
of the Earth’s circumference.
360 10
1
1
dist   2 r    2  4000   2513.274mi
10
10
 2
period  
40.c.) A  5, D  0 (axis of the wave), C  0
2 B
B4
starts at min   cos x
Equation: y  5cos 4 x