Download Name: Math 2412 Activity 2(Due by Feb. 28) Find the quadratic

Name:_____________________ Math 2412 Activity 2(Due by Feb. 28)
Find the quadratic function that satisfies the given conditions. Show your work!
1. The graph has a vertex at 5,12  and it passes through the point 7,10  .
2. The graph passes through the points 0,0, 1,0 , and 2,9  .
3. The graph passes through the points 0,2, 1,0 , and 3,2  .
Sketch the graph of the following polynomial functions.
4. f  x   48 x 2  3x 4
6. f  x   x3  x 2  x  1
5. f  x   3  x  1  x  2 
3
2
7. a) Use the Intermediate Value Theorem to show that the polynomial function
f  x   3x 4  4 x3  3 has at least one zero in the interval 0,1 .
b) Use the Bisection Method to approximate a zero in the interval 0,1 . Complete the
table:
1
2
3
4
Left endpoint (sign)
0(-)
½ (-)
Guess(midpoint) (sign)
½ (-)
Right endpoint (sign)
1(+)
1(+)
Error Bound
½
¼
8. Find the largest area of a rectangle inscribed under the graph of the function f  x   1  12 x in
the first quadrant by doing the following:
a) Express the area of the rectangle as a function of x.
b) What is the domain of the area function?
c) Find the maximum value of the area function.
Find the vertex, and determine if it’s a maximum or minimum.
9. f  x   3 x  2   12
2
10. f  x   2  x  4   8
2
11. f  x   3x 2  12 x  1
Graph the following quadratic functions.
12. f  x   x 2  2 x  15
13. f  x   5  4 x  x 2
14. f  x   3x 2  2 x  4
Write an equation for the quadratic function whose graph satisfies the conditions.
15. Vertex:  3, 4  and passes through 1,4  .
16. Passes through the points  0, 1 , 1, 3 , and  3,5 .
17. The x-intercepts are 1 and 3, and the y-intercept is 4.
18. The x-intercepts are 1 and 3, and the maximum value is 6.
19. A rain gutter is made from sheets of aluminum that are 12 inches wide by turning up the
edges to form right angles. Determine the depth of the gutter that will maximize its crosssectional area and allow the greatest amount of water to flow.
x
12  2x
x
20. A rancher has 1000 feet of fence to construct six corrals, as shown in the figure. Find the
dimensions that maximize the enclosed area.
y
x
21. There is an alternative way to multiply two polynomials called the method of detached
coefficients. It is very similar to the way we multiply numbers. As an example, let’s find
the product of the polynomials x3  3x 2  2 x  4 and 2 x2  x  6 using this method:
1
3
2
4
2
1
6
6 18 12 24
1
3 2
2 6 4
2 7
4
8
5 24
8 24
From the bottom row, we see that the product is 2 x5  7 x4  5x3  24 x2  8x  24 .
a) Multiply the polynomials 4 x3  2 x 2  7 x  1 and 2 x2  x  6 using the method of
detached coefficients.
b) Evaluate each of the two polynomials and their product at x  10 .
c) Compare the by-hand multiplication of 4271 and 216 to the detached multiplication table
in part a).
Determine the end behavior of the following polynomials.
22. f  x   x3  4 x 2
23. f  x   11x 4  6 x 2  x  3
The graph to
The graph to
The graph to
The graph to
the left goes
the right goes
the left goes
the right goes
24. f  x   3 x  1  x  2 
3
2
25. f  x    x 2  2 x  15
The graph to
The graph to
The graph to
The graph to
the left goes
the right goes
the left goes
the right goes
Sketch the graphs of the following polynomial functions. Label the zeros and the y-intercept.
26. f  x  
1
2
 x  5 x  2 
5
27. f  x   
1
3
 x  12   x  4
2
28. f  x   x3  4 x 2  4 x
29. f  x   x3  5 x 2  3x  15
30. f  x    x 4  4 x 2
31. f  x   x 4  6 x3  9 x 2
32. f  x   
1
2
 x  4   x 2  25
16
33. f  x    x3  x  1  x  3
2
2
Find a polynomial function of smallest degree with leading coefficient of 1 that satisfies the
given conditions.
34. Its only zeros are 4,0,3 . The graph crosses the x-axis at 4,0,3 ; lies above the x-axis
between 4 and 0 ; lies below the x-axis between 0 and 3.
35. Its only zeros are 0 and 2. The graph touches the x-axis at 0 and crosses the x-axis at 2; lies
above the x-axis between 0 and 2.
Divide using long division.
36.  x 2  3x  10    x  2 
37.  x3  2 x 2  5 x  6    x  3
x  2 x2  3x  10
x  3 x3  2 x 2  5 x  6
38.  6 x3  17 x 2  27 x  20    3 x  4 
3x  4 6 x3  17 x 2  27 x  20
39.  6 x3  13x 2  11x  15   3 x 2  x  3 
3x2  x  3 6 x3  13x 2  11x  15
Divide using synthetic division.
40.  2 x 2  x  10    x  2 
2 2
1
41.  3x 2  7 x  20    x  5 
10
42.  4 x3  3x 2  3x  1   x  2 
5 3
43.
7
x
2
20
 5 x  5 x3  x 4    5  x 
Use synthetic division to find the indicated function value.
44. f  x   2 x3  11x 2  7 x  5; f  4 
45. f  x   x 4  5 x3  5 x 2  5 x  6; f  3
46. Use synthetic division to divide f  x   x3  2 x 2  x  2 by x  1. Use the result to find all
the zeros of f .
47. Solve the equation 3x3  7 x 2  22 x  8  0 given that 
1
is a solution.
3
48. Find k so that x  1 is a factor of x3  2 x 2  3x  k .
49. Find k so that 4 x  3 is a factor of 20 x3  23x 2  10 x  k .
50. When 2 x 2  7 x  9 is divided by a polynomial the quotient is 2 x  3 and the remainder is 3.
Find the polynomial.
Sketch the graphs of the following rational functions. Indicate intercepts and horizontal and
vertical asymptotes.
51. f  x  
3x
x 1
52. f  x  
4x
x 1
53. f  x  
4 x2
x2  9
54. f  x  
x2  4 x  3
2
 x  1
2
Sketch the graphs of the following rational functions. Indicate intercepts and oblique and
vertical asymptotes.
x2  4
55. f  x  
x
Find a rational function f  x  
x3  1
56. f  x   2
x 9
p x
, in which the degrees of p and q are as small as possible,
q x
that satisfies the given conditions.(57-58)
57. has a vertical asymptote of x  3 , a horizontal asymptote of y  0 , y-intercept of 1, and
no x-intercept.
58. has a vertical asymptote of x  1 , a slant asymptote of y  x , y-intercept of 2 , and xintercepts of 1 and 2 .
Solve the following inequalities by first completing the sign charts. (59-70)
59.  x  3 x  5  0
3
60. x 2  x  6  0
5
3
2
61. 3x 2  16 x  5
5
62. x3  2 x 2  4 x  8  0

1
3
2
2
64.  x  1 x  2  x  3  0
63. x3  4 x2
0
4
3
2
1
65.
 x  3 x  2  0
x 1
3
67.
66.
1
3
1
2
1
2

x 1 x 1
68.
1
1
x
2
x 1
2
x 2  3x  2
0
x2  2 x  3
1
1
2
3
69.  x  2   0
70.  x  2   1
2
2
 
 
 
Inequalities of the form something   something else can be solved by squaring both sides to
 
 
get  something 
2
 
 
2
 
   something else  , which simplifies into
 
 
 
 
2  
2
 something     something else  .
 
 
inequalities: (71 and 72)
71.
2
5

2x  1 x  1
Use this idea to solve the following absolute value
72.
3
1

3x  2 x  2
73. Find the smallest value of the positive constant m that will make mx  1 
1 mx 2  x  1
positive values of x. {Hint: Consider that mx  1  
.}
x
x
1
 0 for all
x
74. If p  x   an x n  an1 x n1 
 a1x  a0 is a polynomial and it has at least n  1 zeros, where
n  0 , then what must be the values of a0 , , an ? Remember, a polynomial of degree n
has at most n zeros(counting repeated zeros).
75. If p  x  and q  x  are polynomials of degree n , where n  0 , and p  x   q  x  for n  1
values, then what can you conclude about p  x  and q  x  ? Consider the previous problem.
76. Determine all polynomials P  x  so that P  x 2  1   P  x    1 and P  0   0 . Find the
2


values of P 1 , P  2  , P  5  , P  26  , P  262  1 , P  262  1  1
2
, and use the previous
problem. What if P  0   1 ?
77. One of the two lines that pass through  3,0  and are tangent to the parabola y  x 2 is the xaxis. Find the other line.
78. A fly is crawling from left to right along the top of the parabola y  7  x 2 . A spider waits
at the point  4,0  . Find the distance between them when they first see each other.
79. Find the values of k so that f  x   x3  4 x 2  2 x  k has a zero between 2 and 1.
80. Find values of a and b so that the polynomial  ax  b   x 5  1   5 x  1 is divisible by
x 2  1.
81. For all positive real numbers a and b, show that
82. If x  0 , show that x 
2ab
ab
.
 ab 
ab
2
1
 2.
x
{Hint: Multiplying both sides by x, leads to x 2  1  2 x . Now manipulate it.}
1 1
83. If x, y  0 , show that     x  y   4 .
x y
1 1
y x
y 1
{Hint: Multiplying out the left side leads to     x  y   1    1  2   y .}
x y
x x
x y
1 1 1
84. If x, y, z  0 , show that      x  y  z   9 .
x y z
1 1 1 1 
85. If x, y, z, w  0 , show that       x  y  z  w   16 .
 x y z w
1 1
86. What about   
 x1 x2

1
  x1  x2 
xn 
 xn  for x1 , x2 ,
, xn  0 ?
87. Suppose that a rational function has vertical asymptotes of x  2 and x  3 only, has zeros
of 1 and 4 only, and has a horizontal asymptote of y  12 .
a) Sketch the graph of such a function.
b) Find a formula for such a function.
88. Suppose that a rational function has a vertical asymptote of x  2 only and a slant
asymptote of y   x  3 .
a) Sketch the graph of such a function.
b) Find a formula for such a function.
Find partial fraction decompositions of the following.
89.
90.
91.
92.
1
x2  4
2x2  3
x  x  1
2
1
x  x 2  x  1
3x 4  x3  20 x 2  3x  31
 x  1  x 2  4 
2
x5  2
93. 2
x 1
94. If 2 x2  3x  c  0 for all real x, then what must be true about c?
95. If 3x 2  bx  7  0 for all real x, then what must be true about b?
96. If ax2  5x  4  0 for all real x, then what must be true about a?
97. Show that
2 3 
3
2

1
2
.
98. Show that
2 3 
99. Show that
9  4 5  5  2.
3
2

1
2
.
23  8 7  4  7 .
100. Show that
101. Show that if x and y are positive numbers, then
102. Show that if 0  x  y , then
x y  x  y.
y  x  yx.




103. Find an integer m so that  3  2 5

104. Find an integer m so that  5  2 3

2
2
 m  is an integer.

2
 m  is an integer.

2
a c
a ac c
 , show that 
 . Use this result
b d
b bd d
1
1
to find 5 different fractions between and .
3
2
105. For a, b, c, and d positive numbers with
Find the domain of the following functions:
106. f  x   x 2  8 x  15
107. f  x  
x2
2x  6
Find the domain and range of the following functions:
108. f  x  
4 x  20
3x  18
x2  1
109. f  x  
x 1
110. f  x  
1
 x  1
2
111. Consider the rational function f  x  
1
.
x  x  1
a) Find the partial fraction decomposition of f  x  .
b) Use the partial fraction decomposition from part a) to find the exact value of the sum
1
1
1
1
.


 
1 2 2  3 3  4
1,000,000 1,000,001
112. Consider the rational function f  x  
2
.
x  x  1 x  2 
a) Find the partial fraction decomposition of f  x  .
b) Use the partial fraction decomposition from part a) to find the exact value of the sum
1
1
1
1
.


 
1 2  3 2  3  4 3  4  5
1,000,000 1,000,001 1,000,002
113. Consider the function f  x   1  1x   1  x11   1  x1 2  
1
 1  x999,998
.
a) Express the function f as a rational function.
b) Use the result of part a) to find the exact value of
1  12   1  13   1  14  
114. Is it possible to have
1
 1  1,000,000
.
1 1
2
? If so, for what values of a and b?
 
a b ab
115. Find a cubic polynomial function p  x  so that p  x  1  p  x   3x 2 .