Download Final Review – Fall Semester Be sure to show all work for full credit. 2.

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33
Algebra 2
Final Review – Fall Semester
Name: ____________________________
Be sure to show all work for full credit.
1. Find the point(s) of intersection algebraically:
y  2 x  8
x  2 y  9
2. Given the function f(x) = 2x2 – 5x + 3 , find
a) f(x) = 3
b) f(x) = 0
y  x 2  6x  5
3. Graph the equations 
x  y  5
c)
f(x) = 0
.
Find the slope, y-intercept, and x-intercept of the equation –x + y = 5.

slope = ___________ y-int: ____________ x-int: ___________
Verify the points of intersection algebraically.
y  x 2  6x  5

x  y  5

points of intersection:
(
,
)
(
,
)
4. Find the equation of the parabola with vertex at (1, 4) that passes through (5, 1).
stretch: ____________
equation: ________________________________
5. Find the equation of the line that passes through the points (4, 5) and (16, -19).
6. Solve:
x  3 3x 1

5
6
x x
 6
3 5


7. Find the points of intersection of the parabolas algebraically.
y  x 2  2x  2

2
y  x  4 x  2

points of intersection: (
,
)
(
,
)
x
10
8. Solve for x:
 x  6 4x  7

2
3
9. Solve for x and y:
y  3x  4
5x  2 y  3
10. Find the equation of the line
that passes through the points
(3, 2) and (27, -38)
11. Solve for x algebraically, be sure to show all work for full credit. Do not use your graphing calculator.
5x2  0.5x 1  0
12. Use the parabola y = x2 + 8x – 24 to answer the following questions:
a) Put the equation in graphing form by completing b) Give the coordinates of the vertex.
the square.
c) Find the x-intercept(s) algebraically.
d) Find the y-intercept(s) algebraically.
13. A ball is dropped from 200 inches, and is allowed to bounce on a firm, level surface. If after the third bounce, the ball rises to a
height of 43.2 inches.
a)
To what height did the ball rise after the first and second bounces?
b) Find the equation that represents the rebound height as a function of which bounce it is.
c)
Find the rebound height of the 10th bounce
d) If the same ball was dropped from 100 feet to what height would it rebound?
14.
Multiply and simplify:
( x 1)(2x  3)2
15. Solve for x:
16
X
16. Simplify:
(3x4 y3 )3 (2 xy 2 )
18x7 y9
17. Solve for x:
4x
3
 4
x3 x
LCD = _______________
18. Sam and James were doing their homework when Sam got to a problem that only had the numbers 5, 2… written down. Sam said
that he remembered the teacher saying this was an arithmetic sequence while James was sure it was a geometric sequence.
a) If Sam was correct, find the next three terms of the sequence and its equation.
b) If James was correct, find the next three terms of the sequence and its equation.
c) James found in his notes that the number 464 is an output of the sequence. Help them end their dispute by determining if it is
Arithmetic, Geometric, both or neither. Justify your answer mathematically.
19. Given a sequence t(n), where t(3) = 40 and t(6) = 31, find the equation if it is:
arithmetic:
geometric:
20. Find the domain and range of the following:
a)
b)
Domain: _________________________
Domain: _________________________
Range: __________________________
Range: __________________________
21. Simplify each of the following WITHOUT A CALCULATOR. You must show enough work to justify
that you did it without a calculator.
a)
(4 xy 2 ) 2 (2x 1 y 5 )
(2x 3 y 2 ) 3
3
1  2
b)  
4 

x
y
f (x) 
16
3x 1
1 
  
8 
x


22. Investigate the function
c)
1
. The graph should include at least 6 plotted points.
x 2

x-int(s): __________________
y-int(s): __________________
asymptote(s): _______________________________
Domain: ___________________________________
Range: _____________________________________
23. You have just inherited $8,000 and want to invest it into either an account that pays 9% annual percentage rate compounded yearly
or an account with 8.75% annual rate compounded quarterly. Each account will be compounded for 5 years. Where do you put the
money? You must justify your answer mathematically and show all work in order to receive full credit.
9% account compounded annually
8.75% account compounded quarterly
sentence: __________________________________________________________________________________________________.
24. Solve each of the equations for x. Show your work and do not use a calculator.
3x
a) 8
 32x2
25. Given
8

b)
2
3
x
1
5
3
. There are three steps required to evaluate
STEP 1 – Fraction form:

8

2
3
without a calculator. Do each of the three steps below.

STEP 2 – Radical form:
STEP 3 – Simplify:
26. a) Simplify:
(2 xy 2 )3 (xy)
3x 6 y 5
b) Simplify and write with no negative exponents:
(a2b3)(ab)1
27. Given t (3)  1250 and t (4)  3125 ,
a) Find the equation if t (n) is arithmetic.
b) Find the equation if t (n) is geometric.
28.
Find the equation in graphing form of the cubic that passes through (0, 8) and has a locator point at (-2, 3).
stretch factor: __________
equation: ___________________________
29.
Without making a table or using the graphing calculator,
sketch an accurate graph of:
y = -2|x + 4| – 1
30.
Without making a table or using the graphing calculator,
sketch an accurate graph of:
x = (y + 3)2 – 1
locator point:
locator point:
stretch:
stretch:
31. Change y = 2x2 + 8x – 24 into graphing form by
completing the square and state the vertex and stretch coefficient.
32. Change x2 + 4x + y2 + 2y = -1 into graphing form by
completing the square and state the vertex and stretch coefficient.
graphing form: ____________________________________
graphing form: _______________________________________
vertex: ________________
vertex: _________________
stretch: ________________
radius: ________________
33. Darren was tossing a ball and he noticed that the path of the ball was a parabola, so he quickly
took measurements. The highest point the ball reached was 5 feet and it landed 16 feet from where he
was standing. Sketch a graph and write an equation that describes the path of the ball.
stretch: ___________
equation: _____________________________________
Solutions:
1. (-5, 2)
2. x = 0, x = 2.5
x = 1, x = 1.5
f(-2) = 21
5. y = -2x + 13
4. stretch = -3/16
equation: y = -3/16(x – 1)2 + 4
7. (-1, -1)
(-2, -2)
10. y = -5/3x + 7
19. arithmetic: t(n) = -3n + 49
geometric: t(n) = 51.61(0.9185)n
20. D: -3 < x < 1
R: 0 < y < 4
radical:
17. x = 1
D: -∞ < x < ∞
R: -4 < y < ∞
23. 9%: y = $12,308.99
8.75%; y = $12,332. 32
1
26.
2
3
b2
a3
3
82
1
simplify:

3
64
28. stretch: 5/8
 equation: y = 5/8(x + 2)2 + 3
31. y = 2(x + 2)2 – 32
vertex: (-2, -32)
stretch: 2
x = 4  2 10
(0, -24)
15. x = 7.512

22. x-int: none
y-int: (0, 0.5)
asymptotes: x = -2, y = 0
D: -∞ < x < ∞; x ≠ -2
R: D: -∞ < y < ∞; y ≠ 0
8
12. y = (x + 4)2 – 40
(-4, -40)
11. x = 2/5, x = -1/2
14. 4x3 – 8x2 – 3x + 9
1
6. x = 23/9
x = 90/8
x = 38.637
9. (1, -1)
8. x = 32/11
13. t(1) = 120, t(2) = 72
t(n) = 200(0.6)n
t(10) = 1.209
60 feet
16. 3x6y2
25. fraction:
3. slope = 1, y-int = 5, x-int = -5
(-5, 0) and (0, 5)
8y 2
3x 2
18. arithmetic: t(n) = -3n + 8
geometric: t(n) = 12.5(0.4)n
arithmetic; n = -152
21.
4 y15
x8
8
x = -4/15

24. x = -5/7
x = 243
27. arithmetic: t(n) = 1875n – 4375
geometric: t(n) = 80(2.5)n


29. LP: (-4, -1)
stretch: -2
32. (x + 2)2 + (y + 1)2 = 4
center: (-2, -1)
radius: 2
30. LP: (-1, -3)
stretch: 1
33. stretch: -5/64
equation: y = -5/64(x – 8)2 + 5