Download CP Final Exam Review Sheet

Name: ________________________ Class: ___________________ Date: __________
ID: A
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 minimum or maximum value of f(x) = x 2 − 2x − 6. Then state the domain and range of the function.
5. Find the zeros of the function h (x) = x 2 + 23x + 60 by factoring.
6. Express 8 −84 in terms of i.
7. Find the zeros of f(x) = x 2 + 7x + 9 by using the Quadratic Formula.
8. Subtract. Write the result in the form a + bi.
(5 – 2i) – (6 + 8i)
9. Multiply 6i (4 − 6i) . Write the result in the form a + bi.
10. Simplify
−2 + 2i
.
5 + 3i
11. 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.
12. Find the product (5x − 3)(x 3 − 5x + 2) .
13. Divide by using synthetic division.
(x 2 − 9x + 10) ÷ (x − 2)
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14. Use Pascal’s Triangle to expand the expression (4x + 3) .
15. Factor x 3 + 5x 2 − 9x − 45.
16. Solve x 4 − 3x 3 − x 2 − 27x − 90 = 0 by finding all roots.
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17. Write the simplest polynomial function with zeros 5, –4, and 2 .
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Name: ________________________
ID: A
18. Identify the leading coefficient, degree, and end behavior of the function P(x) = –5x 4 – 6x 2 + 6.
19. Graph g (x) = 4x 3 − 24x + 9 on a calculator, and estimate the local maxima and minima.
20. Let f(x) = 5x 3 + 7x 2 + 4x − 5. Write a function g that reflects f(x) across the y-axis.
21. The table shows the population of endangered tigers from year 0 (when the study began) to year 20. Write a
polynomial function for the data.
Year
Population
0
280
5
437
10
571
15
781
20
1164
22. 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. Graph the function. After how many days will there be fewer than 321
bacteria?
23. Write the exponential equation 2 3 = 8 in logarithmic form.
24. Write the logarithmic equation log 4 16 = 2 in exponential from.
25. Express log 3 6 + log 3 4.5 as a single logarithm. Simplify, if possible.
26. Express log 3 27 −3 as a product. Simplify, if possible.
27. Simplify
2z 3 − 6z 2
. Identify any z-values for which the expression is undefined.
z 2 − 3z
28. Multiply
8x 4 y 2 9xy 2 z 6
⋅
. Assume that all expressions are defined.
3z 3
4y 4
29. Divide
30. Add
5x 3
25
÷ 9 . Assume that all expressions are defined.
2
3x y 3y
x + 9 −8x − 39
+
. Identify any x-values for which the expression is undefined.
x − 2 x2 + x − 6
31. Subtract
2x 2 − 48 x + 6
−
. Identify any x-values for which the expression is undefined.
x 2 − 16 x + 4
2
Name: ________________________
ID: A
−5
x−6
+
x−4
10
. Assume that all expressions are defined.
32. Simplify
x+3
x−4
33. Solve the equation x − 9 = −
18
.
x
34. Graph the piecewise function.
ÔÏÔÔ
ÔÔ 3x − 1 if x < 0
ÔÔ
f(x) = ÔÌÔ 2x if 0 ≤ x < 4
ÔÔ
ÔÔ
ÔÔ 1 − x if x ≥ 4
Ó
35. Given f(x) = 2x 2 + 8x − 4 and g(x) = − 5x + 6, find (f − g)(x).
36. Given f(x) = 4x 2 + 3x − 5 and g(x) = − 2x + 12, find (fg)(x).
37. Given f (x) = x 3 and g (x) = 4x + 3, find g(f(3)).
38. Given f(x) =
x − 2 and g(x) =
6
+ 1, write the composite function g(f(x)) and state its domain.
x−3
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ID: A
Final Exam Review Sheet
Answer Section
1.
2. The parent function is the cubic function, f (x) = x 3 .
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g (x) = (x + 3) represents a horizontal translation of the parent function 3 units to the left.
3. g(x) = −4x + 5
4. The minimum value is –7. D: {all real numbers}; R: {y | y ≥ –7}
5. x = −20 or x = −3
6. 16i 21
−7 ± 13
2
8. –1 – 10i
9. 36 + 24i
7. x =
2
10. − 17 +
8
17
i
11. −7x 5 + 7x 4 + 3x 3 + 12x 2 − 5x + 6
leading coefficient: –7; degree: 5; number of terms: 6; name: quintic polynomial
12. 5x 4 − 3x 3 − 25x 2 + 25x − 6
13. x − 7 +
−4
x−2
14. 256x 4 + 768x 3 + 864x 2 + 432x + 81
15. (x + 5)(x − 3)(x + 3)
16. The solutions are 5, −2, 3i, and −3i.
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17. P (x) = x 3 − 2 x 2 −
39
2
x + 10
18. The leading coefficient is –5. The degree is 4.
As x →−∞, P(x) →–∞ and as x →+∞, P(x) →–∞
19. The local maximum is about 31.627417. The local minimum is about –13.627417.
20. g(x) = −5x 3 + 7x 2 – 4x – 5
21. f(x) ≈ 0.13x 3 − 2.39x 2 + 40x + 280
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ID: A
22. f(x) = 2032(0.85) t
After about 11.3 days, there will be fewer than 321 bacteria.
23. log 2 8 = 3
24.
25.
26.
27.
4 2 = 16
3
–9
2z; z ≠ 3 or 0
28. 6x 5 z 3
xy 8
5
x+6
30.
;
x+3
The expression is undefined at x = −3.
x−6
31.
; The expression is undefined at x = 4 and x = −4.
x−4
29.
x 2 − 10x − 26
10(x + 3)
33. x = 3 or x = 6
34. 1
35. (f − g)(x) = 2x 2 + 13x − 10
32.
36. (fg)(x) = −8x 3 + 42x 2 + 46x − 60
37. g(f(3)) = 111
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38. g(f(x)) =
+ 1 , x ≥ 2, x ≠ 11
x−2 −3
2