2015 SUMMER PACKET FOR STUDENTS EXITING ALGEBRA 2 TRIGONOMETRY HONORS
These exercises serve two purposes. Firstly, to help you keep up the skills you obtained
during the past year. Secondly, to challenge you in order to help to you extend your skills.
Answers as well as hints to selected questions are given in the back of the document.
Quadratic equations and functions
1) Complete the square for
2)
Solve the equation
3)
For what values of a does
4)
5)
.
have two x-intercepts?
Define a quadratic function f having x-intercepts −6, 4 and y-intercept −3. Give your final
.
answer in the standard form of a quadratic f(x) =
Complete the square for the quadratic function
, find its vertex and x
intercepts and then use the appropriate scale to graph it.
1
Equations and inequalities
1)
2)
3)
4)
5)
6)
7)
Polynomials
1) Use synthetic (short) division, in order to divide the two polynomials:
After you do the division write the division algorithm.
2) Use the factor theorem to decide whether the polynomial
3) Factor the polynomial
is a factor of the polynomial
and find all zeros given that
is a
2
factor.
4) State the possible rational zeros of
Then find the rational zeros.
Inverse functions
1) Prove that the following functions are 1-1 and find the inverse functions.
(A)
(B)
(C)
(D)
Exponential and logarithmic functions and equations
1)
2) Write the following expression as a single logarithm
3)
4) Solve the equation
5) Solve the equation
6)
7)
3
8) Solve the equation
Trigonometry
1) Find the exact values of the six trigonometric functions for each angle:
i.
750o
ii.
2) Find all values of θ, if θ is in the interval [
and has the given function value:
i.
ii.
iii.
iv.
v.
vi.
3) Given that
and
, evaluate the following:
i.
ii.
iii.
iv.
v.
vi.
In which Quadrant does the angle
lie?
4) Verify the following identities:
i.
ii.
5) The diagram shows the graph of the function f given by
where A and B are constants and x is measured in radians.
4
The graph includes the points (1, 3) and (5, 3) which are maximum points of the graph.
i.
Write down the values of f (1) and f (5).
ii.
Show that the period of f is 4.
iii.
The point (3, −1) is a minimum point of the graph. Show that A = 2 and find the value of B.
6) Solve the equation
,
.
HINTS AND ASNWERS
Quadratic equations and functions
1) Answer:
2) Hint: First, use square-root property to find the value of
. Answer:
3)
4) Hint: Start by setting
, then find
.
Answer:
5) Answer: Vertex:
intercepts:
or
Equations and inequalities
1) Hint: Factor first. Answer:
2) Answer:
or
or
.
.
5
3) Hint: First write
as
. Then eliminate denominators by
. Then solve for .
multiplying both sides of the equation by
Answer:
4) Answer:
5) Hint: Expand the left-side to
, then bring all terms to one side. You ‘ve got a
quadratic inequality. Factor and solve as usual.
.
Answer:
6) Hint: Set
Answer:
. Solve the quadratic equation to find the value of
or
Then find
.
7) Hint: Bring both fractions to one side, make the denominators equal, combine the fractions
(Do NOT multiply by the denominators, do you understand why?). Then solve the
inequality as usual.
Answer:
or
.
Polynomials
1) Answer: Quotient:
Remainder:
Division algorithm:
2) Answer: Yes, it is.
3) Hint: First divide by
. Answer:
4) Answer: Possible roots:
. The only rational root is .
Inverse functions
Answers: A)
B)
C)
, with
D)
Exponential and logarithmic functions and equations
1) Answer:
6
2) Answer:
3) Answer:
4) Answer:
5) Answer:
, so no solution.
6) Answer:
.
7) Answer:
8) Answer:
Trigonometry
1) Answers: i.
for
is co-terminal with
. So all six trig numbers of
are the same as
.
ii.
2) Answers: i.
,
,
,
ii.
iii.
,
iv.
v.
vi.
3) Answers: i.
v.
ii.
iii.
iv.
vi. Quadrant III.
4) Start from left-hand side and get the right-hand side.
5) i.
ii. period=
6) Hint: Write
iii.
, set
. Answer:
or
iv.
, solve the quadratic equation for
to get
.
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