Nichols School Mathematics Department
Summer Assignment
Warm-up for Foundations of Precalculus
Who should complete this packet? Students who will be taking Foundations of Precalculus in
the fall of 2016.
Due Date: The first day of school
How many of the problems should I do? – ALL OF THEM
How should I organize my work? You should show all work in a separate sheets of graph
paper Keep your materials in a 3-prong folder or report cover.
How will my teacher know that I’ve done the work? –Your teacher will collect your work on
the first day of school. Your teacher may choose to QUIZ or TEST you on this material if he
or she feels it is necessary – BE PREPARED!
How well should I know this material when I return? – You should recognize that you’ve
seen this material before, and you should also be able to answer questions like the ones in this
assignment. If the material is revisited in your next class, it will only be for a brief amount of
time – your teacher will assume that all you need is a quick refresher.
Note from your teachers:
We feel that this summer work will truly help you succeed this year. We understand that
summer is a time for relaxation and fun, but it is imperative that you spend some time before
you return reviewing your materials. This packet is mandatory, and you must treat it as you
would any other extremely important homework assignment. You will be held accountable for
this material. We also highly suggest that you do a bit of it at a time in the weeks leading up to
school – don’t leave it for the last day!!!
Warm-up for Foundations of Precalculus
Instructions:
▪ Complete the problems on graph paper in pencil.
▪ At the top of each page of your work, write your name.
▪ Complete all the problems carefully. Show enough work to indicate your method of
solution.
▪ Make sure your work justifies your answer.
▪ If a problem requires a graph, you may graph on your calculator, but then sketch the graph
on your paper as part of your solution. Please use a ruler.
▪ Several problems on this assignment require you to make a diagram. Use a ruler and make
an accurate sketch.
▪ Keep the packet and your work in a folder.
▪ Place your completed work in a report cover or 3-prong folder .
Remember, your teacher will collect your work on the first day of school! Late work will be
penalized and may NOT be accepted.
You will continue to use your 3-Ring binder for your math class throughout the school year.
What if I get stuck? - You should check out other additional study materials. Consult a standard
algebra textbook. Find a study buddy or a classmate to help you remember the material. Consult the
following websites for hints and examples:
http://coolmath.com/algebra/Algebra1/index.html
http://www.algebra.com/
http://www.brightstorm.com/math
http://www.khanacademy.org Warm-up for Foundations of Precalculus
page 1
1. Mr. Bosch is cutting timbers to outline a rectangular flower bed. He is using two 8-foot pieces
of timber to outline the bed. He will cut each piece as shown below.
a. Write a quadratic equation to represent the situation. Solve the equation graphically and
then by factoring. Show your work. Where should he make his cut if the area of each
flower bed is to be 15 square feet?
b. Do both answers make sense? Why or why not?
c. Where should Mr. Bosch cut the timbers to outline a 12-square foot flower bed? Identify a,
b, and c and use the quadratic formula to solve.
d. Do both answers make sense? Why or why not?
2. a. Write a quadratic function whose graph has its vertex in the second quadrant.
b.
Name the vertex, give the equation of the axis of symmetry, and tell whether the graph
opens up or down. Justify your reasoning.
3. Solve each equation Approximate to the nearest hundredth.
a.
x2  8x  5  0
b. x 2  3.2 x  0.67  0
c. 3 x 2  22 x  17  0
d. 0.5 x 2  3.5 x  2.7
4. The area of a rectangle is 10.63 cm2. If the length is 3.4 cm longer than the width, find the
length of the rectangle to the nearest hundredth of a centimeter.
5. Approximate the real zeros of each function to the nearest tenth.
a.
f ( x)  2 x3  4 x 2  7 x  6
c.
f ( x) 
b.
f ( x)  4 x 4  0.7 x 3  8.1x 2  3
2 5 6 3 13
x  x  x 1
5
7
6
Nichols School Mathematics Department
Warm-up for Foundations of Precalculus
page 2
6. Find the maxima and minima of each function to the nearest hundredth.
a.
f ( x) 
1 3
1
x  5x2  4 x 
2
8
b.
f ( x)  x 4  x 3  15 x 2  3 x  36
7. Sketch the graph of each function.
a.
f ( x)  x3  x  4
b.
f ( x)  x 4  x 2  2
c.
f ( x)  2 x 5  x 4  8 x3  1
d.
f ( x)  0.4 x 2  2.94 x  2.36
8. Graph f ( x)  2 x 3  6 x 2  3 x  1 and g ( x)  3 x 3  2 x 2  3 x  2 on the same axes. Make an
accurate sketch of both graphs. Find the coordinates of the point P where the two functions
intersect. Round to the nearest hundredth. Circle and label this point on your graph.
9. Determine whether each polynomial is factored correctly. If it is correct, sketch the graph. If it
is incorrect, write the correct factorization and sketch the graph.
b. x 3  1   x  1  x 2  x  1
a. 2 x 2  9 x  5   2 x  1 x  5
4
10. The formula for the volume of a sphere is V   r 3 . If the volume of a sphere is 32.17 cm3,
3
find the radius of the sphere to the nearest hundredth.
11. Write a polynomial function f ( x ) of second degree.
a. Describe how you find the inverse of your function, then find the inverse.
b. Tell in your own words how the graph of a function and its inverse are related.
c. Graph the inverse you found in part a. Is it a function? Why or why not?
12. Find the inverse of each function. Sketch the graph of the function and its inverse on the same
axes.
a.
f ( x)  x 5
b.
f ( x)  x  3
c.
Nichols School Mathematics Department
y  2 x3  1
Warm-up for Foundations of Precalculus
page 3
13. a. Graph the function y  2 x . Sketch the graph, label the y-intercept and 2 other points on
the graph.
b. Graph the function y  2 x . Sketch the graph, label the y-intercept and 2 other points on the
graph.
c. How are the two graphs related? What is an alternate way to write the function in part b?
14. Exactly eight years ago, an investor deposited $1700 into an account that pays 5.75% annual
interest, compounded continuously. If there were no additional deposits or withdrawals, what
amount is in the account today? Use the formula A  Pe rt .
15. According to a commonly used rule, the Rule of 72, an estimation of the number of years it will
take to double an investment invested an n% is given by 72  n .
a. Estimate the number of years it would require for $4000 invested at 8% to be worth $8000.
b. Find the number of years it would take for a savings account of $4000 to double if invested
at 8% compounded annually.
c. How long will it take a $4000 savings account to double when invested at 8% compounded
continuously?
d. Is the Rule of 72 a better estimate for the time required to double an investment deposited at
a fixed rate compounded annually or continuously?
e. Find a more accurate rule for estimating the time required to double money invested at a
constant rate compounded continuously.
16. A ship carrying 1000 passengers has the misfortune to be shipwrecked on a small island; the
passengers are never rescued. The natural resources of the island limit the population to 5780.
The population gets closer and closer to this limiting value, but never reaches it. The population
of the island after time t in years is given by the function:
P (t ) 
5780
1  4.78e 0.4t
a. Find the population after 0, 1, 5, 10, and 20 years.
b. How long does it before the population reaches 5000?
c. Graph the function, and show your answer to parts a & b on the graph.
Nichols School Mathematics Department
Warm-up for Foundations of Precalculus
page 4
17. a. Explain the meaning of x in the equation log 2 65  x .
b. Estimate the value of x in part a. Justify your answer.
18. Evaluate:
a. log(.0001)
b. log 2 128
c. ln  e7 
19. The table below show equations in exponential form and in logarithmic form. Complete the
table by filling in the missing entries. The first row has been completed as an example.
Exponential Form
Logarithmic Form
3y  5
y  log 3 5
103  1000
a.
b.
log 2 32  5
c.
log 0.01  2
d.
y 1  4
e.
4 3 
1
64
20. Find:
a. the 24th term of the sequence 4.2, 6.3, 8.4, ….
b.
the first four terms of an arithmetic sequence if a1  15.38 and d  11.26 .
21. The 4th term of an arithmetic sequence is 12, and the 7th term is -8.8. Find the 1st term and the
21st term.
22. Find the first five terms and the twelfth term of the geometric sequence in which a1  1.1
and r  2 .
Nichols School Mathematics Department
Warm-up for Foundations of Precalculus
page 5
23. Find the first five terms of the sequence in which a1  4.2 and an 1  0.9an  1.5 .
24. a. Explain in your own words what is meant by the term arithmetic sequence.
b. Write an arithmetic sequence.
c. Write the formula for the nth term of your arithmetic sequence. Then find the 30th term.
25. a. Explain in your own words what is meant by the term geometric sequence.
b. Write the formula for the nth term of your geometric sequence. Then find the 10th term.
26. Change each radian measure to degree measure.
a.
11
7
b. 8.25
27. Graph each function on your calculator in degree mode. Sketch the graph; use the graph to
determine the period of the function in degrees.
a.
y  sin(2 x )
1 
b. y  cos  x 
2 
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c. sin  x  90 