Be sure before you start this section that you completely understand function notation.
If you need it, the DOS program fknnotat will give you an unlimited number of examples of functions. Click to download. If this doesn't work, go to DOS downloads on the Precalculus I page.
After working the problems below, look over the attached two pages. Memorize everything in bold: that is, memorize the names of the function families, the form of the basic function and the form of the generalized function. Then read the rest of the material and try to connect it with the problems you have done.
The functions y = x, y = x^2, y = 2^x and y = x^p for various values of p are the four basic functions; the functions we get when we shift and stretch these functions make up the four basic function families. These families have names and generalized functions as described in the table below:
Basic Function
Name of Family
General Form
Basic Characteristics of Graph
y = x
linear
y = mx + b
straight line
y = x^2
quadratic
y = a x^2 + b x + c
parabola
y = 2^x
exponential
y = A* 2 ^ (kx) + c
either increasing or decreasing; horizontal asymptote
y = x^p
power
y = A (x-h) ^ p + c
p negative: horizontal and vertical asymptotes y = c and x = h
p even/odd: graph symmetric or antisymmetric
The purpose of the section is to ensure that you understand this scheme, that you see how the basic characteristics of the graphs of these functions come about, and that you understand how to graph families of functions.
Using x = -3, -2, -1, 0, 1, 2, 3, make tables and sketch graphs for the basic linear, quadratic and exponential functions (see table above under 'basic functions'). You should not have to use a calculator to get any of your y values.
Using x = -5, -2, -1, -1/2, -1/10, 1/10, 1/2, 1, 2 and 5 and for positive powers and excluding x = 0 for the negative powers, make tables and sketch graphs for the basic power functions y = x ^ 2, y = x ^ 3, y = x ^ -2 and y = x ^ -3. Recall that x ^ -a = 1 / x^a. Use a calculator as little as possible; it really should be unnecessary.
Be sure you can construct the table and graph of any of these basic functions in less that 1 minute, anytime you are asked.
In terms of your tables explain:
why the graph of y = x is a straight line,
why y = x^2 is symmetric about x = 0 (i.e., taking the same values on either side of x = 0)
why y = 2^x keeps increasing as x increases, and why the graph approaches the x axis for negative values of x
why y = x^3 is antisymmetric about x = 0 (i.e., taking the same values except for the - sign on either side of x = 0)
why y = x^-2 and y = x^-3 rise more and more steeply as x approaches 0, and why their graphs approach the x axis as we move away from the y axis.
You are familiar with the y = a x^2 + b x + c form of the family of quadratic functions. This family consists of all functions whose graphs are parabolas with vertical axes of symmetry (i.e., all parabolas which open straight up or down). It would be very difficult to sketch all such parabolas. But we can look at what happens for certain combinations of a, b and c values.
For example if b and c are both zero, we are left with y = a x^2. As we saw in our analysis of quadratic functions, y = a x^2 is just a vertical stretch of the basic y = x^2 function and has a graph that fits into the scheme of the graphs shown below:
Different families of quadratic functions have different patterns, but they all involve parabolas. A couple of other families are shown below.
The first graph is obtained by letting a = 1, b = 0, and c = -5, -4, ..., 4. So we have the functions y = a x^2 + bx + c = 1 x^2 + 0x + c, or y = x^2 + c, with c varying from -5 to 4. The functions are therefore
y = x^2 -5
y = x^2 -4
y = x^2 -3
...
y = x^2 +4.
The second graph is obtained by letting a=1, c=1 and b = -3, -2, ..., 5. The family is y = ax^2 + bx + c = 1 x^2 + b x + 1, or y = x^2 + bx + 1, with -3 < b < 5. The functions graphed are therefore
y = x^2 - 3x + 1
y = x^2 - 2x + 1
y = x^2 - x + 1
...
y = x^2 + 5x + 1.
In each of these examples we have specified some of the parameters a, b and c, and we have given a range for others.
The exercises that follow apply these ideas to a variety of functions and situations.
2. Explain why the family y = x^2 + c sketched above has a series of identical parabolas, each 1 unit higher than the one below it.
3. Using the figure above, determine the approximate x coordinate of the vertex of every graph in the y = x^2 + bx + 1 family
Use your knowledge of quadratic functions to obtain the x coordinate of the vertex of each of the functions sketched in this figure. How well does this agree with your estimates?
4. Sketch the graph of the exponential family y = A * 2^x for the values A = -3 to 3.
Sketch the graph of the exponential family y = 2^x + c for the values c = -3 to 3.
The general form of the exponential function is y = A * 2^(kx) + c.
How would you describe the parameters A, k and c for the first of these families? How would you describe the same parameters for the second family?
5. Sketch the graph of the power function family y = A (x-h) ^ p + c for each of the following parameter ranges:
for p = -2: A = 1, h = 0, c = -3 to 3.
for p = -3: A = 1, h = -3 to 3, c = 0.
6. Make a table of y vs. x for the basic linear function, using x = -3, -2, -1, 0, 1, 2, 3. Sketch the graph. On the same set of axes sketch the graph of the generalized linear function for m = 2, b = 0, and another for m=1, b=2.
7. If f(x) = 2x + 3, then write expressions for the following: f(-2), f(3), f(x+3), f(x) + 3, 3f(x) and f(3x). All these expressions are different, and all involve substituting the expression in parentheses for x in the original form of the function.
8. Make a table for y vs. x for the basic exponential function y = 2^x and plot its graph below. On the same graph plot the generalized exponential function for A = 2, b = 2 and c = 2. Plot also the generalized exponential function for A = 1, b = .5 and c = 1.
9. If f(x) = 2^x, then write expressions for the following: f(-2), f(3), f(x+3), f(x) + 3, 3f(x) and f(3x). All these expressions are different, and all involve substituting the expression in parentheses for x in the original form of the function.
10. The illumination y from a certain florescent bulb is given as a function of distance x by the generalized power function for p = -1 with A = 370, h = 0 and c = 0. Determine the illumination at distances of 1, 2, 3 and 4 units, and sketch a graph.
The four basic function families are briefly outlined below. Memorize the boldfaced information and don't forget it. Remember that you should be able to construct a table and a graph for any of the basic functions in less than a minute, and you should simply remember how to get the values on the table and construct the graph. You should ideally be able to visualize the table and graph any time you wish.
You already know much of the information given about quadratic functions. You can see that the scheme of description is pretty much the same for all functions; it's just the details that differ from one function to the next.
y = f(x) = mx + b (slope-intercept form)
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
y = f(x) = ax^2 + bx + c (quadratic formula form)
or, alternatively,
y = f(x) = a(x-h)^2 + c (standard form)
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
Sequence behavior:
Sequence f(n) has linear first difference, constant second difference
y = f(x) = A b^x + c
or, alternatively,
y = f(x) = A ( 2 ^ (kx) ) + c
Key parameters:
growth factor b = 1 + growth rate
horizontal asymptote c
initial difference A
for 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
f(x) = A (x-h)^p + c
Key parameters:
vertical shift c (horizontal asymptote for negative powers)
horizontal shift h (vertical asymptote for negative powers)
vertical stretch A
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)
