Download Pre-Calculus H

Pre-Calculus H
Summer Assignment
Part 1: Functions
(0, 4)
(–3,0)
(4, –1)
1. For which of the following functions is
f (3)  f (3) ?
(D)
f ( x)  4 x2
f ( x)  4
4
f ( x) 
x
f ( x)  4  x3
(E)
f ( x)  x4  4
(A)
(B)
(C)
(6,0)
3. The graph of y  f ( x) is shown above.
If 3  x  6 , for how many values of x
does f ( x)  2 ?
(A)
(B)
(C)
(D)
(E)
0
1
2
3
more than 3
y
y

6

4
2
–6 –4 –2
O
2
4
6
x
–2
–4
x
–6
2. The figure above shows the graph of the
function h. Which of the following is
closest to h(5)?
(A)
(B)
(C)
(D)
(E)
1
2
3
4
5
4. The figure above shows the graph of a
quadratic function f that has a minimum at
the point (1, 1). If f (b)  f (3) , which of
the following could be the value of b?
(A)
(B)
(C)
(D)
(E)
–3
–2
–1
1
5
y
 (7, 6)


(–6, 0)

(6, 0)
O
x

5. Based on the graph of the function f
above, what are the values of x for
which f ( x) is negative?
(A)
(B)
(C)
(D)
(E)
6  x  0
0 x6
6 x7
6  0  6
6  x  0 and 6  x  7
7. The quadratic function g is given by
g ( x)  ax2  bx  c , where a and c are
negative constants. Which of the
following could be the graph of g?
(A)
x
O
(B)
x
O
3  2 x2
for all nonzero x, then
x
what is the value of f (2) ?
6. If f ( x) 
(A)
(B)
11
2
7
2
1
2
5
2
(C)

(D)
(E)

7
(C)
x
O
(D)
x
O
(E)
O
x
x
0
1
2
f(x)
a
24
b
8. The table above shows some values for
the function f . If f is a linear
function, what is the value of a  b ?
(A)
(B)
(C)
(D)
(E)
24
36
48
72
It cannot be determined from
the information given.
9. Let the function f be defined by
f ( x)  x  1 . If 2 f ( p)  20 , what is the
value of f (3 p) ?
Questions 10-11 refer to the following
functions g and h.
g (n)  n2  n
h(n)  n2  n
10. g (5)  h(4) 
(A)
(B)
(C)
(D)
(E)
0
8
10
18
32
11. Which of the following is equivalent to
h(m  1) ?
(A)
(B)
(C)
(D)
(E)
g (m)
g (m)  1
g (m)  1
h(m)  1
h(m)  1
y
g
4
3
f
2
1
O
12. If the function f is defined by
f ( x)  x2  bx  c , where b and c are
positive constants, which of the
following could be the graph of f?
(A)
O
x
1
2
3
x
4
13. The graphs of the functions f and g are
lines, as shown above. What is the value
of f (3)  g (3) ?
(A)
(B)
(C)
(D)
(E)
1.5
2
3
4
5.5
(B)
O
x
yx
2
Q
P
yax
(C)
O
x
2
O
14. The figure above shows the graphs of
y  x2 and y  a  x2 for some constant
(D)
a. If the length of PQ is equal to 6,
what is the value of a ?
O
x
(A)
(B)
(C)
(D)
(E)
(E)
O
x
6
9
12
15
18
15. Let the function h be defined by
h(t )  2(t 3  3) . When h(t )  60 , what
is the value of 2  3t ?
(A)
(B)
(C)
(D)
(E)
35
11
7
–7
–11
17. Let the operation  be defined
a  b  aa bb for all numbers a and b
where a  b . If 1  2  2  x , what is
the value of x?
(A)
(B)
(C)
(D)
(E)
4
3
2
1
0
y
B (– 12 , b)
C ( 12 , c)
y
O
4
A (– 12 , a)
x
4
16. The shaded region in the figure above is
bounded by the x-axis, the line x  4 ,
and the graph of y  f ( x) . If the point
(a, b) lies in the shaded region, which of
the following must be true?
I. a  4
II. b  a
III. b  f (a)
(A)
(B)
(C)
(D)
(E)
x
y = f(x)
I only
III only
I and II only
I and III only
I, II, and III
D ( 12 , d)
Note: Figure not drawn to scale.
18. In the figure above, ABCD is a
rectangle. Points A and C lie on the
graph of y  px3 , where p is a constant.
If the area of ABCD is 4, what is the
value of p?
y  f ( x)
y  g ( x)
(–1, 3)
(2, 1)
x
O
x
O
(1, –3)
(4, –5)
19. Let the function f be defined by
f ( x)  2 x  1 . If 12 f ( t )  4 , what is
the value of t?
(A)
(B)
(C)
(D)
(E)
3
2
7
2
9
2
49
4
81
4
(A)
(B)
(C)
(D)
(E)
20. For all positive integers w and y, where
w  y , let the operation  be defined
w y
2
. For how many
2w y
positive integers w is w  1 equal to 4?
by w  y 
(A)
(B)
(C)
(D)
(E)
21. The figures above show the graphs of the
functions f and g. The function f is
defined by f ( x)  x3  4x . The function
g is defined by g ( x)  f ( x  h)  k ,
where h and k are constants. What is the
value of hk?
None
One
Two
Four
More than four
–6
–3
–2
3
6
22. Let f be the function defined by
f ( x)  x2  18 . If m is a positive
number such that f (2m)  2 f (m) , what
is the value of m ?
Pre-Calculus H
Summer Assignment
Part 2: Complex Numbers
1. Expanding Complex Numbers
A. What is
1
1
1

 75 ?
7
0
i
(i  1) i
B. What is (2  3i)5 ?
2. Graphing Complex Numbers
A. Let z1  5  2i . In the space provided below, graph the following complex numbers:
z1 , i  z1 ( z1 ) 2 and z1 .
B. Calculate z1 .
3. Equating Like Coefficients. Find the values of x and y in the following:
A. 2 x  5i  4(2  yi)  12
B. (2  i)2  (2  yi)  x  5i
4. Square Root of a Complex Number
Example: What is the (complex) value of
4  3i ?
Solution: We suppose that the answer is some number. We don’t know what the answer
is, so we’ll call it a  bi , since that can be any number you could think of. In other words,
we need to find a and b such that: 4  3i  a  bi
Now algebra happens. Square both sides and get 4  3i  a 2  b2  2abi and get a system
of two equations to solve by equating like coefficients: a 2  b2  4 and 2ab  3 .
Solving that systems (steps omitted for brevity) yields a  3 22 and b   22 or a   3 2 2
and b 
2
2
. Thus
4  3i  3 2 2 
A. What is the value of
2
2
i or
3  4i ?
4  3i   3 2 2 
2
2
i
B. What is the value of
8  6i ?
5. Proofs (Verifications). Let z  a  bi . Prove:
1
A. Re( z )  ( z  z )
2
B. 4 z
2
 z  z  z  z 
2
2
Pre-Calculus H
Summer Assignment
Part 3: Graphing Polynomials
1. If f ( x)  3x3  8x2  15x  4 has one
root given by f (1)  0 , find the other
roots of f .
3. A polynomial function g ( x) has exactly
three x intercepts, at x  2 , x  1 and at
x  5 . If g (2)  4 , give a possible
equation for g ( x) .
2. If f ( x)  4x3  11x2  4x  20 has one
root given by f (2)  0 , find the other
roots of f .
4. A quadratic polynomial
g ( x)  ax2  bx  c has rational values
for a, b and c. If g (1  i)  0 and
g (1)  2 , find the values of a, b and c.
5. Find the quotient and remainder when
4 x3  2 x  1 is divided by x 2  3 .
6. Find the value of k such that when
x 4  3x3  kx 2  4 x  40 is divided by
x  2 , the remainder is 56.
7. In the space provided below, sketch the
graph of y  ( x  3)( x  1)( x  4) . Your
graph need not be to scale, but it must
correctly indicate the nature of the graph
at its intercepts, turning points, and end
behavior.
8. In the space provided below, sketch the
graph of y  ( x  4)( x 2  x  4) . Your
graph need not be to scale, but it must
correctly indicate the nature of the graph
at its intercepts, turning points, and end
behavior.
Pre-Calculus H
Summer Assignment
Part 4: Logarithms
1. If x  0 and y  0 , then log x2 ( y) 
(A) log x y 2
(B) log x
(D) (log x y)2
(C) log x ( 12 y)
y
2. Which of the following correctly solves for x in the equation, ln  x  2   2 ?
(A) e 2  2
(B) ln 2  e
(D) 2  e 2
(C) 2e 2
3. Which of the following is the solution to the equation 3e2 x  2e x  1 ?
1
(A) x   ln1
(B) x 
(C) x   ln 3
ln 3
(D) x  ln 23
4. Which of the following is the graph of G( x)  log(2  x)
(A)
(B)
(C)
(D)
5. Which of the following is NOT true?
(A) log a ( x  y )  log a x  log a y
(B) ln e x  x
(C) log a 1  0
(D) log10 x 
log x x
log x 10
6. Rewrite the expression as a single logarithm: log 7 2  2log 7 2
(A) ln 8
(B) 1
(C) log 8 7
(D) log 7 8
4
7. Which equation best represents the graph shown?
2
(A)
(B)
(C)
(D)
y  log( x 2 )
y  log1 x
y  log 5 x
y  log( x  1)
-10
-5
5
-2
-4
8. Which of the following is a graph of h( x)  4
x
(A)
(B)
(C)
(D)
9. Identify the domain and range of the function h( x)  7  5x
(A) Domain: (, ) ; Range: (, 7)
(C) Domain: (5, ) ; Range: (7, )
(B) Domain: (, 7) ; Range: (, )
(D) Domain: (5, 5) ; Range: (7, 7)
10. State the asymptote of the function g ( x)  2 x  3
(A) y = 3
(C) y = –2
(B) y = 2
(D) y = –3
11. An object is placed in a hot oven until its temperature becomes 245 F. It is then taken out and
left to cool in a room at a temperature of 72 F. Which of the following equations could
represent the objects temperature as a function of time t, once it has been left out to cool?
(A) T (t )  245  72e0.14t
(B) T (t )  72  173e0.14t
(C) T (t )  72  173e0.14t
(D) T (t )  245  72e0.14t
12. Which of the following is equivalent to e2 x ln a ?
(A) e 2 a x
(B) a x  2
(C) ea
x
(D) e2  a x
10
13. Solve the following. Give exact (calculator ready) answers and if your answers are in terms
of logarithmic functions, use natural log ( ln ).
A. e7 x  34 x 1
B. log3  2 x  1  2  log3  x  9 
C. 2e x  3e  x  5
D. 3ex 2  4e  2 x
14. Population growth is more realistically modeled logistically, rather than exponentially. In an
experiment with the protozoan Paramecium, a biologist determines a system’s carrying
capacity to be 48 (in units of 10,000 Paramecium) and its growth constant to be 0.556. Growth
is measured over days and the population’s initial value is 6.
(A) Write an equation that gives the population of the protozoan at a time t.
(B) What will the population be after 4 days ?
(C) When will the population be 50 ?
Pre-Calculus H
Summer Assignment
Part 5: Conic Sections
1. Find the length of segment AB , and the
coordinates of the midpoint of segment
AB whose endpoints are A( –4, 7) and
B( 6, 1).
4. Calculate the eccentricity of the conic
x2 y 2

1
section given by
36 20
2. Give the equation of a circle with center
(–2, 3) with radius 5.
5. Consider the parabola with vertex
(–2, –1) and focus (–2, 2). Give the
equation of the parabola.
3. A hyperbola has vertices at ( 2, 0) and at
(  2, 0) . Its asymptotes are given by
6. The equations of a hyperbola and a line
are given by x2  2 y 2  7 and x  2 y  1
respectively. Find the coordinates of all
points of intersection of the their graphs.
y  12 x and y   12 x . Give the equation
of the hyperbola.
7. Sketch the region bound by the curves y  9  x 2 and x  2 y  6 . Find the coordinates of
all points of intersection of the curves.
8. Give an expression for A( x) the area of a rectangle as a function of an x coordinate, with one
side on the x-axis and inscribed inside the top half of the ellipse given by y  9  94 x 2 . The
rectangle is symmetric about the y axis.
(x, y)
Pre-Calculus H
Summer Assignment
Part 6: Trigonometry
I. First and foremost, students must know the values of all six primary trigonometric functions of
common angles on the unit circle. This knowledge must be automatic and accurate. Students will
be expected to demonstrate their knowledge of the unit circle on the first day of class under timed
circumstances.
You should search for “the Unit Circle” online for blank worksheet templates to practice with and
work out the values of all six primary trigonometric functions of common angles on [0, 2 ) .
You will be expected to give the values of six randomly selected angles in a 90 second time
period.
Name:
Give the values of the indicated
trigonometric function for each
corresponding highlighted angle.
Date:
Period:
Seat:
 cos 
 sin  
 tan  
π
0
 sec 
 csc 
 cot  
II. Additional Problems:
1. If  is an angle such that cos   0 and
tan   0 , state the quadrant that contains
.
2. For the diagram shown below, give the
value of csc .

(–6, –2)
x
3. If  is an angle in the second quadrant
such that tan    5 , find the value
cos  .
5. If sin 230  k , give another angle 
between 180 and 180 such that
sin   k .
4. If x  3cos  , simplify the expression
6. Simplify:
cot   1
1  tan 
9  x2
in terms of functions of  .
x
7. Simplify: 3sin x  2sin x  2sin x  csc x  6cos x  cos x
8. The arc length of a sector is
radius.

6
. The area of the same sector is

. Find the sector’s angle and
4
1.2
2.00
2.25
1.00
0.75
0.6
2.75
0.50
0.4
0.2
3.00
1.2
3.25
1.25
0.8
2.50
9. In the figure at right, the values along the
circle are angles measured in radians. The
values along the axes are x and y coordinates
respectively. Use the figure to approximate
sin(5.75) to the nearest tenth.
1.50
1.75
-0.8 -0.6 -0.4 -0.2
-0.2
0.25
0.2 0.4 0.6 0.8
-0.4
3.50
-0.6
3.75
5.75
-0.8
4.00
4.25
4.50
5.50
4.75 5.00
1.2
1.2
6.25
6.00
5.25