The Basic Function Families

This section is a more detailed outline of the basic function families than that given in Introduction to Function Families.  You should memorize the scheme used to describe the characteristics of each family.

Everything you should know to date is indicated in bold.  You don't have to know the details of the bolded topics unless that information is also in bold; but you should know the scheme used for each family (i.e., Basic Function, Generalized Function, Key Parameters, Function Family, etc.).  This scheme will help you organize the details you learn in the rest of the course.

Note:  Some of the graphs have been rendered in light colors which will not print well on a black-and-white printer.  Check your printouts against the graphs as shown on this Internet page.

• The Family of Linear Functions

• The family of Quadratic functions

• The family of Exponential functions

• The families of Power functions

The Family of Linear Functions

Basic function: y = f(x) = x

 

Table of the Basic Linear Function y = f(x) = x
plus a vertical stretch and a vertical shift of this function
x	f(x)	3 f(x)	f(x) + 3
-2	-2	-6	1
-1	-1	-3	2
0	0	0	3
1	1	3	4
2	2	6	5

   

Generalized function:

y = f(x) = mx + b (slope-intercept form)

Function family:

Key parameters:

constant slope m

y-intercept b

Key characteristics of graph:

Straight line, never vertical

Constant slope

Key points for graphing:

y-intercept (0,b)

x-intercept (-b/m)

point 1 unit to right of y-intercept (1,b+m)

Typical situations:

Pendulum: Force vs. displacement: Required force vs. displacement of pendulum

Flow: Horizontal range of stream vs. time for flow from side of vertical uniform cylinder

Income: Money earned vs. hours worked

Demand: Demand for a product vs. selling price (simplified economic model)

Straight-line approximation to any continuously changing quantity over a short time interval

Rate equation:

dy / dt = c

Difference equation:

a(n+1) = a(n) + c

Sequence behavior:

Sequence f(n) has constant first difference

The family of Quadratic functions

Basic function:  y = x^2

 

Table of the Basic Quadratic Function y = f(x) = x^2
plus vertical stretch 2 f(x) = 2 x^2
and vertical shift f(x) + 2 = x^2 + 2
x	f(x)	2 f(x)	f(x) + 2
-4	16	32	18
-3	9	18	11
-2	4	8	6
-1	1	2	3
0	0	0	2
1	1	2	3
2	4	8	6
3	9	18	11
4	16	32	18

 

Generalized function:

y = f(x) = ax^2 + bx + c (quadratic formula form)

or, alternatively,

y = f(x) = a(x-h)^2 + c (standard form)

Function family:

Key parameters:

Vertical stretch factor a (both forms)

Horizontal shift h (standard form f(x) = a(x-h)^2 + c); also x coordinate of vertex

Vertical shift c (only for standard form f(x) = a(x-h)^2 + c); also y coordinate of vertex for this form only

Key characteristics of graph:

Shape of a vertical parabola

Symmetry with vertical axis through vertex

Slope of graph changes at a constant rate

Key points for graphing:

Vertex:     xVertex = -b/2a,     yVertex = f (xVertex)

Zero(s): x = [ -b +- `sqrt(b^2-4ac))] / (2a), whenever b^2-4ac >= 0

1 unit right and left of vertex: (xVertex+1, yVertex+a) and (xVertex-1,yVertex+a)

Typical situations:

Depth vs. time for flow from a uniform cylinder

Altitude of a thrown ball vs. time

Revenues vs. selling price (simplified economic model)

Curvature-based approximation to any continuously changing quantity over a short time interval

Rate equation:

d [dy / dt] /dt = constant (rate of change changes at a constant rate)

dy / dt = k `sqrt(y)

Difference equation:

a(n+1) = a(n) + c n

Sequence behavior:

Sequence f(n) has linear first difference, constant second difference

The family of Exponential functions

Basic function: y = f(x) = 2^x

 

Table of the Basic Exponential Function y = f(x) = 2^x
plus vertical stretch y = 2 f(x)
and vertical shift y = f(x) + 2
x	f(x)	2 f(x)	f(x) + 2
-3	0.125	0.25	2.125
-2	0.25	0.5	2.25
-1	0.5	1	2.5
0	1	2	3
1	2	4	4
2	4	8	6
3	8	16	10
4	16	32	18

Generalized function:

y = f(x) = A b^x + c

or, alternatively

y = f(x) = A ( 2 ^ (kx) ) + c.

Function family:

Key parameters:

growth factor b = 1 + growth rate

horizontal asymptote c

initial difference A

alternative form:  k, where growth factor b = 2^k so k = log(b) / log(2)

Key characteristics of graph:

Either always increasing or always decreasing

Rate of increase or decrease is always constantly increasing or constantly decreasing

Approaches asymptote y = c either as x large either in the positive or the negative direction (either to the far right or the far left of the graph), depending on whether b > 1 or b < 1.

Key points for graphing:

y-intercept (0, A + c)

1 unit to right and to left of y-intercept: (1, A b + c) and (-1, A/b + c)

Typical situations:

Compound interest: Value of investment vs. time

Unrestricted population growth: Population vs. time

Temperature approach to room temperature: Temperature vs. time

Radioactive decay: Amount remaining vs. time

Rate equation

dy / dt = k y

Difference equation

a(n+1) = k (a(n) - c), a(1) = A

Sequence behavior:

Sequence f(n) has constant ratio of first differences

The families of Power functions

Basic function: y = f(x) = x^p

 

Table of the Basic Power 3 Function y = x^3
plus vertical stretch y = 2 f(x)
and vertical shift y = f(x) + 2
x	f(x)	2 f(x)	f(x) + 2
-3	-27	-54	-25
-2	-8	-16	-6
-1	-1	-2	1
0	0	0	2
1	1	2	3
2	8	16	10
3	27	54	29
4	64	128	66

Generalized function:

f(x) = A (x-h)^p + c

Function family:

Key parameters:

vertical shift c (horizontal asymptote for negative powers)

horizontal shift h (vertical asymptote for negative powers)

vertical stretch A

Key characteristics of graph:

(h, c) is intersection of vertical and horizontal asymptotes for negative powers

(h, c) is extreme point for even positive powers

(h, c) is point of inflection, where curvature changes from positive to negative, for odd powers p.

Positive powers: for x > h, increases at increasing rate if p>1, increases at decreasing rate if p<1.

Negative powers: symmetric about x = h is p even, antisymmetric if p odd.

Key points for graphing y = A x^p:

x = 0: (0, A) for positive power, x = 0 is vertical asymptote for negative power

.5, 1 and 2 units to right of x=h: (.5, A (.5)^p), (1, A), (2, A (2^p))

y changes from A(.5)^p to A to A(2^p)

positive powers: 3 points strictly increasing if A positive

p>1: increasing rate of increase, increasing slope, more so for greater power p

p<1: decreasing rate of increase, decreasing slope, more so for lesser power p

negative powers: 3 points strictly decreasing if A positive

All p < 0: decreasing rate of decrease, increasing slope, less so for p nearer to 0

1 unit to left of x=h: (h - 1, A + c) for even power, (h - 1, -A + c) for odd power, showing antisymmetry

 

 

Typical situations:

Pendulum: period of pendulum vs. length (power .5)

Pendulum: frequency of pendulum vs. length (power -.5)

Area: surface area vs. scale for a family of geometrically similar objects (power 2)

Volume: volume vs. scale for a family of geometrically similar objects (power 3)

Strength: strength vs. weight for geometrically and physiologically similar individuals (power 2/3)

Illumination: illumination vs. distance from a point source (power -2)

Illumination: illumination vs. distance from a line source (power -1)

Rate equation:

Beyond the scope of this course

Difference equation:

Uncommonly encountered; beyond the scope of this course

Sequence behavior:

Sequence f(n) is a power function sequence of power p-1