A2H Midterm Review

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Algebra II Honors - Midterm Exam Review
**All questions are NO CALCULATOR, unless specified.**
Chapter 1
1. For the function y = –x2 – 6x – 7, find the
vertex and axis of symmetry.
Ⓐvertex (3, – 2); axis of symmetry x = 3
Ⓑvertex (–3, 2); axis of symmetry x = 4
Ⓒvertex (–3, 2); axis of symmetry x = –3
Ⓓvertex (3, –2); axis of symmetry x = –4
2. If the graph of y = ax2 + bx + c opens
down, which of the following must be true?
Ⓐa < 0
Ⓑa > 0
Ⓒc < 0
Ⓓc > 0
3. Which function does not have a maximum
value?
Ⓐy = –x2 – 5x – 6
Ⓑy = –x2 – x – 6
Ⓒy = 3x2 – 15x + 2
Ⓓy = 49 – x2
4. What is the vertex of y = –3(x – 2)2 – 4?
Ⓐ(–2, –4)
Ⓑ(–2, 4)
Ⓒ(2, –4)
Ⓓ(2, 4)
5. What are the x–intercepts of
y = –2(x – 7)(x + 2)?
Ⓐ–7 and 2
Ⓑ7 and –2
Ⓒ14 and –4
Ⓓ14 and –2
6. Factor the expression m2 – 4m – 21.
7. Which value of c makes the expression
x2 – 5x + c a perfect square trinomial?
5
5
Ⓐ
Ⓑ
2
2
25
Ⓒ
Ⓓ25
4
9. Factor the expression 8x2 + 28x + 12.
Ⓐ2(x + 2)(4x + 3)
Ⓑ2(x + 3)(2x + 1)
Ⓒ4(x + 1)(2x + 3)
Ⓓ4(x + 3)(2x + 1)
Ⓐ(m – 7)(m – 3) Ⓑ(m – 7)(m + 3)
Ⓒ(m + 7)(m – 3) Ⓓ(m + 7)(m + 3)
8. What are the roots of the equation
z2 + 11z – 42 = 0?
Ⓐ–3, –14
Ⓑ3, –14
Ⓒ–3, 14
Ⓓ3, 14
10. Simplify the expression
Ⓐ
4
13
Ⓒ4 2 3
2
2 3
Ⓑ2
Ⓓ
42 3
5
11. What are the solutions of the equation
w2 = –9w?
Ⓐ–9, 3
Ⓑ0, –9
Ⓒ0, 9
Ⓓ1, 9
12. What are the solutions of
23 = 2(x – 3)2 + 7?
Ⓐ 3
Ⓑ 3 2 2
13. What are the solutions of –3 – y2 = 24?
Ⓐ 3 3
Ⓑ 3i 3
Ⓒ 9 3
Ⓓ 9i 3
14. What is the standard form of the
i
expression
?
2i
1
2
Ⓐ i 1
Ⓑ i 1
2
3
Ⓒ 2 2
Ⓓ6  4 2
i
2
Ⓒ 1
15. What is the vertex of y = 3x2 –30x + 77?
Ⓐ(–5, –2)
Ⓒ(5, –2)
Ⓑ(–5, 2)
Ⓓ(5, 2)
1
5
16. What is the value of c if the discrimant of
–2x2 – 3x + c is 41?
Ⓐ–4
Ⓒ
17. How many real number solutions
does the equation 7x2 – 5x + 1 = 0 have?
2
5
Ⓓ i
11
2
Ⓑ4
Ⓓ
25
4
18. A rectangular garden is 25 feet long by
10 feet wide. You have enough mulch to
cover 1000 square feet.
a. You would like to extend both the
length and the width of the garden by
x feet to use up all of the mulch. Write
an equation to represent the area of
the new garden.
b. Solve the equation from part (a).
c. Which solution do you have to reject?
Explain.
19. Graph the function y = x2 – 2x + 1. Label the
vertex and the axis of symmetry.
20. Tell whether the function y = x2 – 5x + 6 has a
minimum value or a maximum value. Then find
that value.
21. Graph the function y = 2(x – 1)2. Label the
vertex and the axis of symmetry.
22. Graph the function y = –(x + 2)(x – 2). Label
the vertex, the axis of symmetry and the x–
intercepts.
23. Write the quadratic function
y = 2(x + 3)(x – 1) in standard form.
24. Write the quadratic function
y = 5(x – 2)2 – 5 in standard form.
25. Determine which number sets each quantity 26. State the domain and the range of each
belongs to.
function in interval notation.
3
d. p
a. 4.35
c. b. 49
4
27. State the domain and the range of each
function in interval notation.
28. Write the equation of the parabola with the
given vertex and point.
vertex (-3, 1) passing through (2, -4)
Chapter 2
1. What is the simplified form of
2ab 2 c 2
3
2a
Ⓒ 2
3b
Ⓐ
8a 2bc 1
?
12ab3c
2a
3b 2 c 2
2a
Ⓓ
3bc
Ⓑ
3. Which equation is the graph of the
polynomial function shown?
2. What is (3.2 × 105)(1.4 × 10–2) written in
scientific notation?
Ⓐ4.48 × 103
Ⓑ4.48 × 107
Ⓒ44.8 × 103
Ⓓ44.8 × 107
4. What is the degree of the polynomial
h(t) = –8t2 + 5 – 3t3?
Ⓐ1
Ⓑ2
Ⓒ3
Ⓓ4
Ⓐf(x) = –2x3 + x2 – 2
Ⓑf(x) = 3x4 – x2 + 1
Ⓒf(x) = x3 – x + 7
Ⓓf(x) = –2x4 + x2 – 1
5. What is the greatest common monomial
factor of 9x3y2 + 15x2y – 6xy2?
Ⓐ3x2
Ⓑ3y2
Ⓒ3xy
Ⓓ3x2y2
7. What is the complete factorization of
3x4 – 3x2?
Ⓐ3x2(x2 – 1)
Ⓑ3x2(x – 1)(x + 1)
Ⓒ3x(x – 1)(x + 1)
Ⓓ3(x4 – x2)
6. Given f(x) = 3x – 5 evaluate f-1(0).
8. If x + 3 is a factor of x3 – x2 – 17x – 15,
what is another factor?
Ⓐx + 1
Ⓑx – 1
Ⓒx + 5
Ⓓx – 3
9. If x – 2 is a factor of a polynomial f(x),
which of the following statements does
not have to be true?
Ⓐf(2) = 0
Ⓑf(–2) = 0
Ⓒ2 is a root of f(x).
Ⓓ2 is a zero of f(x)
10. Which is not a possible rational solution of
f(x) = 3x3 – 11x2 + 5x – 6?
1
2
Ⓒ 2
Ⓐ
2
3
Ⓓ 6
Ⓑ
11. How many zeros does 0 = –7m3 – m4 + 1 have?
12. Use direct substitution to evaluate
2x3 – 4x2 + 8x – 3 for x = –2
13. Use synthetic substitution to evaluate
4x4 – 2x3 – 3x2 + 3x for x = 2.
14. Graph f(x) = –x4 + 1.
15. Perform the indicated operation.
(x + 4)2 (x – 2)
16. Perform the indicated operation.
(x3 + 2x – 1) – (2x2 +4x – 2)
17. Factor the polynomial completely using any
method.
x3 + 3x2 + x + 3
18. Divide.
(x3 – 4x2 – 2x + 3) ÷ (x + 1)
19. Find all real zeros of f(x) = x3 – 7x – 6.
20. Graph the function f(x) = (x + 1)2(x + 4).
21. Identify the end behavior for each of the
following.
a. f(x) = -3x4 + 3x2 + 1
22. Which polynomial represents the volume of
the cone shown?
b. f(x) = 5x(3x – 1)2
2 3
20
x – x2 – 4x +
3
3
20 2  x 10
x 

Ⓑ
3
3
3
2
20
 x
Ⓒ
4x +
3
3
Ⓐ
Ⓓ
Chapter 3
50
4 x3
 4x2 – 5x –
3
3
1. What is the value of (–243)3/5?
Ⓐ–27
Ⓑ–3
Ⓒ3
Ⓓ27
2. What is the solution to 3x5 + 350 = –379?
729
Ⓐ 5
Ⓑ–3
3
729
Ⓒ3
Ⓓ5
3
3. Which expression is the simplest form of
4 3 32  3 32
4. What is the simplified expression of the
length of the triangle’s hypotenuse?
Ⓐ 33 4
Ⓑ 63 4
Ⓒ6
Ⓓ 16 3 2  4
3x1/2
2x3/2
5. What is the simplified form of
 z 2 16 z 3  3 36 z 7 ?
Ⓐ  z3 z
Ⓑ 14z3 z
Ⓐ 2 x3 2  3x 1 2
Ⓑ2x3/2 + 3x1/2
Ⓒ 4 x3  9 x
Ⓓ4x3 + 9x2
6. If h(t) = t2/3 – 9 and j(t) = 3t + 5t2/3,
what is h(t) – j(t)?
Ⓐ–4t2/3 – 3t – 9
Ⓑ4t2/3 + 3t + 9
Ⓒ3t + 6t4/3
Ⓓ–7t7/3 – 9
Ⓒ 14z 4 z Ⓓ 92z3 z
7. What is g(f(x)) if f(x) = 3x2 and
g(x) = 2x1/2?
Ⓐx 6
Ⓒ6 x
Ⓑ2 x 3
Ⓒx ≥
Ⓓ6x
9. Which function represents the inverse of the
graph shown?
1
5
Ⓐy = –5x + 3
Ⓑy = x – 3
1
5
Ⓓy = 5x + 3
Ⓒy = x + 3
8. Given u(x) = 4 x  1 and v(x) = x – 5
what is the domain of u(v(x))?
ⒶAll real numbers Ⓑx ≥ 0
1
4
Ⓓx ≥
21
4
10. What is the inverse of the power function
8 3
g(t) = –
t?
27
2
8
Ⓐh(t) = – 3 t
Ⓑh(t) = – 3 t
3
27
33
3
Ⓒh(t) = – t
Ⓓh(t) = – t
2
2
11. Which of the following pairs of functions are
not inverses of one another?
Ⓐu(x) = x – 2; v(x) = x + 2
1
1
Ⓑu(x) = 5x – 1; v(x) = x +
5
5
Ⓒu(x) = x3 + 1; v(x) = 3 x-1
Ⓓu(x) = x  2 ; v(x) = x2 + 2
14. What is (are) the solution(s) to
x – 2 = 2x 1 ?
Ⓐx = 1
Ⓑx = 5
Ⓒx = 1 and 5
ⒹNo solution
13. What are the domain and range of the
function y = 5 x  2 ?
ⒶDomain: all real numbers;
range: all real numbers
ⒷDomain: x ≥ 2; range: all real numbers
ⒸDomain: all real numbers; range: y ≥ 0
ⒹDomain: x ≥ 2; range: y ≥ 0
16. Evaluate –274/3 without using a calculator.
15. Solve (x – 5)2/3 – 2 = 2?
17. Verify that f and g are inverse functions.
f(x) = 2x + 5, g(x) =
18. Find the inverse of the function.
x 5
2
f(x) =
19. Graph the function. Then state the domain
and range.
y=2
x2 – 2
3
x+2=
23. Solve the equation.
4x  2
4x4 - 3x2 + 3x -1
x2 - x +1
28  x
24. Let f(x) = 2x3 – 5 and g(x) = 3x2. Perform the
indicated operation and state the domain.
f(g(x))
25. Identify the remainder.
a.
13
x 3 – 1
2
22. Solve the equation.
2x  8
3x  5 =
2x  5
3
20. Graph the function. Then state the domain
and range.
y=
21. Solve the equation.
4=
12. The graph of y = x is shifted 2 units up
and 3 units to the left. Which is the equation
of the translated function?
Ⓐy = x  2 – 3
Ⓑy = x  2 –3
Ⓒy = x  2 + 3
Ⓓy = x  3 + 2
b.
(3x
5
- 4x3 + 2x - 5) ( x+1)
-1
Chapter 4
1. Which function is shown in the graph?
Ⓐf(x) = 2(2.3)x – 2
Ⓑf(x) = 4(2.3)x
Ⓒf(x) = 4(2.3)x + 2
Ⓓf(x) = 5(2.3)x – 3
3. Which function represents exponential
growth?
2
Ⓐu(t) = –7.0  
3
t
2
14e x
1
Ⓐ e5 x
2
1 2
Ⓒ e9 x – x
2
4. What is the horizontal asymptote of the
function y = 2(0.3)x – 1 – 4?
t
3
Ⓑu(t) = –7.0  
2
Ⓒu(t) = 7.0(0.8)t
-t
æ 10 ö
Ⓓu(t) = 7.0 ç ÷
è9ø
5. What is the simplified expression of
7  e3 x 
2. Gasoline costs $1.99 per gallon. If the price
per gallon increases an average of 6% per
month, which function models the
exponential growth of the pricing?
Ⓐf(x) = 1.06(1.99)x
Ⓑf(x) = 1.99(1.06)x
Ⓒf(x) = [1.06(1.99)]x
1.99
Ⓓf(x) =
1.06 x
?
1
2
7
Ⓓ e8 x
2
Ⓑ e8 x
7. Which expression is equivalent to x?
Ⓐlog x
Ⓑlog 2x
x
Ⓒlog 10
Ⓓlog 10x
Ⓐy = – 4
Ⓒy = 2
Ⓑy = 0.3
Ⓓy = 4
6. Which function does not model
exponential decay?
3
Ⓐr(x) = e –3 x
4
4
Ⓑr(x) = e –3 x
3
Ⓒr(x) = 4e–3x
3
Ⓓr(x) = e3 x
4
8. What is an equivalent expression for
2 log4 3 + log4 2?
Ⓐ2 log4 6
Ⓑlog4 6
Ⓒlog4 12
Ⓓlog4 18
9. Which of the following is not equivalent to
log5 8?
ln 8
ln 5
Ⓒ3 log5 2
Ⓐ
Ⓑ2 log5 4
Ⓓlog5 4 + log5 2
11. What is the solution to the equation
log4 4x + 2 log4 x = 4?
Ⓐ1
Ⓑ2
Ⓒ3
Ⓓ4
13. What is the value of x in the equation
 2 x –10 
1
x
?
3 = 
9
15. (Calculator) Your grandparents deposited
$2000 into a college savings account for you 5
years ago. If the account pays 2.5% annual
interest, compounded quarterly, find the
current balance of the savings account.
17. State the domain and range y = e–3x.
19. Expand the expression.
a. In 16x2
ln x
2 ln 8
ln 8
Ⓒy =
2 ln x
Ⓐy =
2 ln x
ln 8
2 ln 8
Ⓓy =
ln x
Ⓑy =
12. A pheasant farmer started her farm with 120
pheasants. An analysis of her records shows
that her pheasant population has increased
by 15% each year. The farmer wants to
determine a model of pheasant population
growth using an exponential function.
According to her model, what will the
pheasant population be in 10 years?
Ⓐ311
Ⓑ485
Ⓒ501
Ⓓ1380
14. Graph the function y = 2 • 3x + 1 – 2. State the
domain and range.
16. (Calculator) You buy a computer for $1200.
The value of the computer decreases by 30%
each year. Find the value of the computer after
4 years.
18. Evaluate the logarithm without using a
calculator.
a. log5 25
b. log1/3 81
c. 10log5x
d. log4 16x
20. Condense the expression.
a. log3 2x + 3 log3 4x
b. In 72x – 2 In 2y
2 x3
b. log5
4y
22. State the domain and range of
21. Solve.
1
a. 27(2x + 4) =  
9
10. What is the inverse of the function
y = 82x?
( x – 46)
b. log2 (x + 8) = 4
c. log6 x + log6 (x + 16) = 2
y = log3 (x + 2) – 2.
ANSWERS
Ch 1
1. C
2. A
3. C
4. C
5. B
6. B
7. C
8. B
9. D
10. C
11. B
12. B
13. B
14. D
15. D
16. B
17. 0
18. a. 1000 = (10 + x)(25 + x)
b. x = 15 or x = –50 c. x = –50 because you
cannot have a negative length.
19.
Ch 2
1. B
2. A
3. A
4. C
5. C
6.
5
3
7. B
8. A
9. B
10. A
11. 4
12. –51
13. 42
14.
15. x3 + 6x2 – 32
16. x3 – 2x2 – 2x + 1
17. (x + 3)(x2 + 1)
18. x2 – 5x + 3
19. –2, –1, 3
20.
20. minimum;
21.
-1/ 4
21.
a)
x ® ¥, f ( x) ® -¥
x ® -¥, f ( x) ® -¥
22.
22. A
23. y = 2x2 + 4x – 6
24. y = 5x2 – 20x + 15
25.
a) R, Q
b) R, Q, Z, W,
N, D
26. D: -¥,¥ R: -4, 4
(
) [ ]
27. D: ( -¥,3] R: ( -¥, 2]
28. y= -
1
(x + 3)2 + 1
5
c) R, Q
d) R, I, T
b)
x ® ¥, f ( x) ® ¥
x ® -¥, f ( x) ® -¥
Ch 3
1. A
2. B
3. B
4. C
5. B
6. A
7. B
8. D
9. C
10. C
11. D
12. D
13. D
14. B
15. 13, -3
16. –81
17. f(g(x)) = x, g(f(x)) = x
18. f–1(x) =
19.
Ch 4
1. C
2. B
3. B
4. A
5. A
6. D
7. C
8. D
9. B
10. A
11. D
12. B
13. 4
14.
domain: all reals;
range: y > –2
3x  5
2
domain: x ≥ –2;
range: y ≥ –2
15. $2265.42
16. $288.12
17.
domain: all reals;
range: y > 0
20.
domain: all real numbers;
range: all real numbers
21. 36
22. 3
23. 7
24. 54x6 – 5; all real numbers
18. a. 2
b. -4
c. 5x
d. 2x
19. a. In 16 + 2ln x
b. 3log5 x + log5 2 – log5 y– log5 4
20. a. log3 128x4
b. ln
21. a. 10
22.
c. 2
b. 8
domain: x>-2;
range: all reals
18x
y2
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