Mathematics Practices

Answer all of these questions on separate paper. You may
draw graphs on this sheet where specified.

1. Let Plot the graph
of f. Sketch the result here.

2. By synthetic substitution , show that 2 is a zero of
.

3. Write as the product of a linear binomial and a
cubic polynomial . By synthetic substitution, show
that -1 is a zero of the cubic polynomial.

4. Write as the product of two linear binomials and
a quadratic trinomial . Show that the quadratic
trinomial has no real zeros .

5. Find the two nonreal complex zeros of

6. Explain how the graph in Problem 1 agrees with the
zeros you found algebraically in Problems 2–5.

7. Let Plot the graph
of g on the same screen as in Problem 1. Use the
result to find the two integer zeros of

8. To five decimal places, find the other two real zeros
of g(x).

 

For Problems 9–14, a freight train backs up, stops, and
then goes forward. The displacement, kilometers, of
its engine from a railroad crossing is given by

where t is time, in minutes (see graph).

9. What is the average velocity of the train from
to minutes? What is the average velocity
from to   minutes? Based on these
numerical answers , what would you conjecture is the
instantaneous velocity at ?

10. Write an expression for the average velocity of the
train from 3 minutes to t minutes. By simplifying the
resulting fraction , calculate algebraically the
instantaneous velocity at minutes. How does
the answer compare with your conjecture in
Problem 9?

11. Plot a line through the point where on the
previous graph , with slope equal to the
instantaneous velocity at time How does the
line relate to the graph?

12. At time minute, is the train approaching the
railroad crossing or moving away from it? At what
instantaneous rate?

13. Let Find and .Show that
these values agree with the instantaneous velocity of
the train at and ?

14. in Problem 13 is the velocity function. At what
two times is the velocity equal to zero? At the
positive value of t, is the displacement positive or
negative? Give a real-world interpretation of your
answer.

15. What did you learn as a result of doing this
Exploration that you did not know before?

1. Mr. X, the math teacher, is making up a test for his
students. He wants them to find the particular
equation of a cubic function, but all he will reveal
about it is that for its three zeros, and , this
is true:

Find the particular equation for . Plot the graph,
and sketch the result here.

2. One of the zeros of is a real number. Are the
other two zeros real numbers or nonreal complex
numbers? How can you tell this from the graph?

3. Confirm your conclusion in Problem 2 by calculating
the three zeros.

4. Use the answers to Problem 3 to show that the zeros
have the sum and product properties specified in
Problem 1.

5. Function has a double zero
(a zero of multiplicity 2) because two of the factors
are zero for the same value of x, namely, .
Plot the graph and sketch the result here. What
happens to the graph when there is a real zero of
multiplicity 2?

6. Let . By synthetic
substitution, show that -1 is a zero of .Then
find the other zeros of .

7. A quartic function such as is supposed to have
exactly four zeros. How do you explain the answer to
Problem 6 in regard to this property?

9. At a double zero, the graph of a polynomial function
just touches the x-axis but does not cross it. Is the
same thing true of a triple zero? How can you tell
from the graph that a polynomial function might
have a triple zero?

For Problems 10–13, a bumblebee flies past a flower. It
decides to go back for another look. Just before it reaches
the flower, it turns and flies off. The bumblebee’s
displacement from the flower, feet, is given by

where t is time, in seconds (see graph).

10. What is its average velocity from to
seconds?

11. Write an expression for the bee’s average velocity of
the train from 6 seconds to t seconds. By simplifying
the resulting fraction, calculate algebraically the
bee’s exact instantaneous velocity at seconds.

12. Plot a line through the point where on the given
graph, with slope equal to the instantaneous velocity
at time . How does the line relate to the graph?

13. At what (positive) time was the bee closest to the
flower? How close?

14. What did you learn as a result of doing this
Exploration that you did not know before?

1. Find the particular equation of a polynomial function.

a. From points, e.g., a cubic function containing the
points (2, 7), (3, 34), (4, 91), and (5, 190).

b. From zeros, e.g., a cubic function with leading
coefficient 1 and zeros -5, 3, and 6. Write in
factored form and then multiply and simplify .
Confirm by graphing.

c. From sums and products of zeros, e.g., sum= 1,
product = -15, sum of pairwise products = -7.

2. Find zeros of a polynomial function.

a. By graphing, e.g.,



Explain how you know that two of the zeros of
are complex numbers.

b. By synthetic substitution, e.g.,



Find the complex zeros using the quadratic formula.

3. Find discontinuities in a rational function .

a. Vertical asymptotes, e.g.,

Sketch the graph. Explain how the graph is a
transformation of

b. Removable discontinuities, e.g.,



Find the x- and y-coordinates algebraically.
Confirm graphically using a friendly window that
includes the discontinuity as a grid point.

4. Find the rate of change of a polynomial function.

a. Average rate numerically, e.g., the displacement
of a moving object, , in feet, is given by
where x is in seconds. Find
the average rate of change of for the time
intervals [3, 3.1], [3, 3.01], and [3, 3.001]. Give the
units.

b. Instantaneous rate numerically, e.g., use the
results of part a to make a conjecture about the
instantaneous rate of change of at .
Explain the basis for your conjecture.

c. Instantaneous rate algebraically, e.g., for in
part a, write the equation for a rational algebraic
function that gives the average velocity of the
moving object for the time interval [3, x]. Simplify
the equation by doing the calculations needed to

remove the removable discontinuity. Use the
simplified equation to calculate the instantaneous
velocity at exactly.

d. Instantaneous rate by shortcut, e.g., let
. Show that equals the
instantaneous rate of change of at
Explain how you can find the equation for
from the equation for .

e. Instantaneous rate graphically, e.g., the figure here
shows the graph of from part a. Plot a line on
the graph at the point with slope equal to the
instantaneous rate of change of at that point.
How is the line related to the graph?

The next figure shows the graph of a function.
Estimate graphically the instantaneous rate of
change of the function at .

f. Instantaneous rate verbally, e.g., “The
instantaneous rate at is the —?— of the
average rates as —?—” and “The instantaneous
rate of change of at a particular point is
called the —?— of ”

5. What did you learn as a result of doing this
Exploration that you did not know before?