Math 135
Midterm
Review Sheet
1. Solve the equation: x
4

3
 x
3

3
 x
2
10

9
2. Solve the equation:
3. Find all real solutions to the equation: x
2 
2( x

3) x
2 
3 x
 
0
For problems 4, define your variable, set up an appropriate equation and solve. Label your answer.
4. A square plot of land has a building 60 feet long and 40 feet wide at one corner. The rest of the land outside the building forms a parking lot has area 12,000 square feet, what are the dimensions of the entire plot of land?
5. Use the graph to determine the x- and y-intercepts and determine the domain of the function.
Evaluate the function at the given value:
6. ( )
 x
2 
3 x

1; f ( 2)
Use the vertical line test to determine if the graph represents a function.
7.
Find all real solutions of the equation.
8.

9. x 3
2

5 x
1
3

36
10. Find an equation for the line with the given properties. Express your answer using either the slope-intercept form.
Containing the points (-3, 5) and (1, -2)
11. Find the slope and the y-intercept of the line. Graph the line.
3 x

7 y

21
12. The equation of a line L is given. Find the slope of the line parallel and perpendicular to L.
3 x

7 y

21
13. Find an equation for the line with the given properties. Express your answer using the slope-intercept form.
Perpendicular to 3 x

7 y

21 and contains (2, -1)
14. Find an equation for the line with the given properties. Express your answer using the slope-intercept form.
Parallel to 5 y

20 and contains (-2, 3)
Sketch the graph of each function. Be sure the label key points. Find the domain. Locate any intercepts, and based on the graph find the range.
15.

3
2
 x
2 x

1 if x if 1
 
1 x if x

1
1 16.
 x

3
3 if x if x


0
0
Solve the polynomial.
17. 2 x
3  x
2 
18 x

9

0
18. Solve by completing the square.
2 x
2 
12 x
 
0
Solve and give the restrictions for the following:
19. x
1

1
 x
1

1
 x 2
2 x

1
20. Find the average rate of change for f x

2 x
2 
12 x

17 from x = 1 to x = 2

Graph each function using y
 x
2 and using the techniques of shifting, compressing, stretching, and/or reflecting.
21. y

1
2 x
2 
3 22. f x

2 x
2 
12 x

17
Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. Determine the domain and the range of the function. Determine where the function is increasing and where it is decreasing.
23. f x

3 x 2 
6 x 24 f x

3 x
2 
6 x

2
Determine which functions are polynomials. For those that are, state the degree. For those that are not, tell why not.
25.

1

5 x
3
26. g x

7 x
3   x

0.7
Use transformations of the graph of y
 x
4
or y
 x
5
to graph each function.
27.
 
2( x

3)
4 
5 28. f x
 
( x

3)
5
For each polynomial function, list each real zero and its multiplicity; determine whether the graph crosses or touches the x-axis at each x-intercept; and find the power function that the graph of f resembles for large values of x.
29.
  x
 4
2( 3) ( x
2 
5)
3 30. f x

5
2 x x
2 
7)
For each function find the x- and y-intercepts; determine whether the graph of f crosses or touches the x-axis at each xintercept; determine the end behavior; determine the maximum number of turning points; use the x-intercepts to find the intervals on which the graph of f is above the x-axis, and plot the points obtained in this process to connect with a smooth continuous curve.
31. f x

2
2 x x

4)( x

2) 32.
 
3
2 x x

1)( x

2)
2
Find the domain of each rational function.
33.
 x
2 
3 x

4 x 2 
4 x
2 x

5 x

6
2 
9
Find the vertical, horizontal, and oblique asymptotes, if any, for each rational function.
35.

3 x 2
2 x

3 x

5
2 
1
36.

3 x 3   x
2 
1
2
Analyze the graph of each function.
37.
 x

1 x
2  
6
38.

34. x
2   x
2 
9
12

39. You have 120 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
40. Among all pairs of numbers whose sum is 30, find the two that gives the greatest product.

Solve the inequality.
41.

1

20 42.
3 x

5 x

2

2
Solutions: (work in progress)
1. x
4

3
 x
3

3
 x
2
10

9
4
 x

3
 
3
 x

3
 
10
4 x

12

3 x 9 10
7 x
 
10
7 x

7 x

1
2. x
2 
3 x
 
0 x

   
 
2    

3

2
5
3. x
2 
2( x

3) x
2 
2 x
 
0 x

   
2(1)
2    

2

2


2
 
2
20

 i 5
2
4.

60 40
 
12000
 x
2
2400

12000
 x
2
14400
 x
2
120
 x
120 ft X 120 ft i 5

5. x-intercept: (3,0); y-intercept (0,3); Domain: all reals
6. 11 7. No 8.. yes
9.
2 1 1 x 3

5 x 3

36 let
  x 3
5 36
5 36

0
  
9
  
4
 
0
 
9
  
4 x 3
1

9 x

729 x 3
1
 
4 x
 
64
 
3, 5
    m

5 ( 2)

3 1

7
4
10. y
 mx
 b

7
 b so b
4


1
4 y


7 x

1
4 4
11.
3 x

7 y

21 m


3
; y
 int : (0, 3)
7

12.
3 x

7 y

21 parallel

7
3
: ;
7 perpendicular :
3
13.
3 x

7 y

21 perpendicular :
7
3 y
 mx
 b
7
 b so b

3

17
3 y

7

3 x

17
3
14. y = 3
15.
17.
2 x
3  x
2 
18 x

9

0 x
2

2 x

1
   x

1
 
0

2 x

1
  x
2 
9


0 x
 1
2 x
 
3
16.

18.
2 x
2 
12 x
 
0 x
2 
6 x

5
2 x
2 
6 x

9

5

2
9
 x

3

2 
23
2 x
23
2 x
23
2 x

3

2 23
2
19. x
1

1
 x
1

1
 x
2
2 x

1 x restrictions
1
 x

1
 
1
 x

1
 
2 x
2

2 x
1
 x no solution

20.
( )

2 x
2 f
   f
 

2

1
12 x

17

2(2) 2    
17 


2

1
2(1) 2    
17
21.

49

31

18
1
22.
23. opens up; Vertex(-1,-3); Axis of Symmetry: x = -1; x-int: (-2,0) and (0, 0); y-int: (0, 0); domain: ; Range: [3, )

24. opens up; Vertex(-1,-1); Axis of Symmetry: x = -1; x-int:
Domain: ; Range: [ 1, )
3
3
; y-int: (0, 2);
25. Yes; 3
27.
26. Yes; 3
28.
29. x-intercepts x = -3 (multiplicity 4 and touches) y
 
2 x
10

30. x-intercepts x = 0 (multiplicity 2 and touches) x
 
7 (both multiplicity 1 and crosses)
31. x-intercepts: x = 0 (touches); x=-4 (crosses); x = 2 (crosses)
y-intercept: y = 0 End behavior: y

2 x
4
Maximum turning points: 3 Above x-axis:

, 4
  
32. x-intercepts: x = 0 (touches); x=-1 (crosses); x = 2 (touches)
y-intercept: y = 0 End behavior: y
 
3 x
5
Maximum turning points: 4 Above x-axis:

, 1

33.
34.
~
 
35. VA:
36. VA: x
 
2
2
HA: y

3
2 x
 
1 OA: y = 3x

37. x-intercepts: x = -1; y-intercept: 0,

1
6
Crosses HA at x = -1
VA: x = 3; x = -2 HA: y = 0
38. x-intercepts: x = 4; y-intercept:

0, 4 3

VA: x = 3 HA: y = 1 Hole at x = -3
39.
A
 w (60
 w )
A
  w
2 
60 w h


2 b a

2

60
 

30; so width = 30 and Area = 900

40. first number x and second number is y so x + y = 30
P
 xy
 x

30
 x
   x 2 
30 x h

 b
2 a

2

30
 

15; so x = 15 and so y = 30 - 15

15 so the maximum product is 15 times 15 = 225
41. (-5, 4)
42. ( , 2]

[9,

)