Name: ________________________ Class: ___________________ Date: __________
ID: A
Algebra II Final Exam Review Sheet
1. Translate the point (2, –3) left 2 units and up 3 units. Give the coordinates of the translated point.
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2. Identify the parent function for g (x) = (x + 3) and describe what transformation of the parent function it
represents.
3. Let g(x) be the transformation, vertical translation 3 units down, of f(x) = −4x + 8. Write the rule for g(x).
4. Find the zeros of the function h (x) = x 2 + 23x + 60 by factoring.
5. Find the zeros of f(x) = x 2 + 7x + 9 by using the Quadratic Formula.
6. Subtract. Write the result in the form a + bi.
(5 – 2i) – (6 + 8i)
7. Multiply 6i (4 − 6i) . Write the result in the form a + bi.
8. Simplify
−2 + 2i
.
5 + 3i
9. Rewrite the polynomial 12x2 + 6 – 7x5 + 3x3 + 7x4 – 5x in standard form. Then, identify the leading
coefficient, degree, and number of terms. Name the polynomial.
10. Find the product (5x − 3)(x 3 − 5x + 2) .
11. Divide by using synthetic division.
(x 2 − 9x + 10) ÷ (x − 2)
12. Factor x 3 + 5x 2 − 9x − 45.
13. Solve the polynomial equation 3x 5 + 6x 4 − 72x 3 = 0 by factoring.
14. Identify the leading coefficient, degree, and end behavior of the function P(x) = –5x 4 – 6x 2 + 6.
15. For f (x) = x 3 + 1, write the rule for g (x) = f (x) + 2.
16. Let f(x) = 5x 3 + 7x 2 + 4x − 5. Write a function g that reflects f(x) across the y-axis.
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Name: ________________________
ID: A
17. Use finite differences to determine the degree of the polynomial that best describes the data.
x
y
–3
–12
–1
–7
1
–21
3
–51
5
–93
7
–142
18. A bacteria population starts at 2,032 and decreases at about 15% per day. Write a function representing the
number of bacteria present each day. After seven days how many bacteria will be present?
19. Write the exponential equation 2 3 = 8 in logarithmic form.
20. Write the logarithmic equation log 4 16 = 2 in exponential from.
21. Express log 3 6 + log 3 4.5 as a single logarithm. Simplify, if possible.
22. Express log 2 64 − log 2 4 as a single logarithm. Simplify, if possible.
23. Simplify the expression log 4 64.
24. Distance that sound travels through air d varies directly as time t, and d = 1,675 ft when t = 5 s. Find t when
d = 5,025 ft.
25. Determine whether the data set represents a direct variation, an inverse variation, or neither.
x
2
3
4
y
420
280
210
26. Given: y varies directly as x, and y = −5 when x = 2.5. Write and graph the direct variation function.
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Name: ________________________
ID: A
27. Multiply
8x 4 y 2 9xy 2 z 6
⋅
. Assume that all expressions are defined.
3z 3
4y 4
28. Simplify
6r 2 − 12r
. Identify any r-values for which the expression is undefined.
r −2
29. Divide
25
5x 8
÷
. Assume that all expressions are defined.
6x 7 y 6y 3
6
30. Solve the equation x + 5 = .
x
31.
x
x+4
+3=
4
x−2
ÏÔÔ
ÔÔ −7 if x < 4
32. Graph the piecewise function h (x) = ÌÔ
.
ÔÔ 7 if x ≥ 4
Ó
3
Name: ________________________
ID: A
33. Graph the piecewise function.
ÏÔÔ
ÔÔÔ 3x − 1 if x < 0
ÔÔ
f(x) = ÔÌÔ 2x if 0 ≤ x < 4
ÔÔ
ÔÔ
ÔÔ 1 − x if x ≥ 4
Ó
34. Given f(x) = 2x 2 + 8x − 4 and g(x) = − 5x + 6, find (f − g)(x).
35. Given f(x) = 4x 2 + 3x − 5 and g(x) = − 2x + 12, find (fg)(x).
36. Given f (x) = x 3 and g (x) = 4x + 3, find g(f(3)).
37. Given f(x) = x − 2 and g(x) =
6
+ 1, write the composite function g(f(x)) and state its domain.
x−3
4
ID: A
Algebra II Final Exam Review Sheet
Answer Section
1.
2. The parent function is the cubic function, f (x) = x 3 .
3
g (x) = (x + 3) represents a horizontal translation of the parent function 3 units to the left.
3. g(x) = −4x + 5
4. x = −20 or x = −3
−7 ± 13
2
6. –1 – 10i
7. 36 + 24i
5. x =
2
8. − 17 +
8
17
i
9. −7x 5 + 7x 4 + 3x 3 + 12x 2 − 5x + 6
leading coefficient: –7; degree: 5; number of terms: 6; name: quintic polynomial
10. 5x 4 − 3x 3 − 25x 2 + 25x − 6
11. x − 7 +
−4
x−2
12. (x + 5)(x − 3)(x + 3)
13. The roots are 0, –6, and 4.
14. The leading coefficient is –5. The degree is 4.
As x →−∞, P(x) →–∞ and as x →+∞, P(x) →–∞
1
ID: A
15. To graph g (x) = f (x) + 2, translate the graph of f (x) up 2 units.
16. g(x) = −5x 3 + 7x 2 – 4x – 5
17. The fourth differences are constant. A quartic polynomial best describes the data.
18. f(x) = 2032(0.85) t
After about 11.3 days, there will be fewer than 321 bacteria.
19. log 2 8 = 3
20.
21.
22.
23.
24.
25.
4 2 = 16
3
4
3
15 sec
Inverse variation
2
ID: A
26. y = –2x
27. 6x 5 z 3
28. 6r; r ≠ 2
xy 2
29.
5
30. x = −6 or x = 1
31. 3
32.
33. 1
34. (f − g)(x) = 2x 2 + 13x − 10
35. (fg)(x) = −8x 3 + 42x 2 + 46x − 60
36. g(f(3)) = 111
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37. g(f(x)) =
+ 1 , x ≥ 2, x ≠ 11
x−2 −3
3