precalcs qc part 1 and key

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Page 1: PreCalcs QC part 1 and Key

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Page 14: PreCalcs QC part 1 and Key

Please use the space below to write your response(s) to the writing assignment provided by your

teacher. If there are multiple tasks to the question, please clearly label the number or letter of each

task in the column to the left of your answers. If you need additional pages for your response, your teacher can provide them.

Please write the name of the writing assignment here: _____________________________________

Task

Page 15: PreCalcs QC part 1 and Key

QualityCore® Reference SheetPrecalculus

Triangles

Law of Sines = =

Law of Cosines a 2 = b 2 + c 2 − 2bc cos A

Area of a Triangle Area = bc sin A

Conic Sections

Circle (x − h)2 + (y − k)2 = r 2

Parabola, y = a(x − h)2 + k

opening vertically

Parabola,x = a(y − k)2 + h

opening horizontally

Ellipse,major axis horizontal

Ellipse,major axis vertical

Area of an ellipse A = πab

Hyperbola,transverse axis horizontal

Hyperbola,transverse axis vertical

Sequences and Series

Arithmetic Sequence an = a1 + (n − 1)d

Arithmetic Series sn = (a1 + an)

Geometric Sequence an = a1�r n − 1�

Finite Geometric Series sn = where r ≠ 1

Infinite Geometric Series s = where ⎪r⎪ < 1

Exponential Functions

Discretely Compounded Interest

ContinuouslyCompounded Interest

Discrete, ContinuousExponential Growth

a1_____1 − r

a1 − a1rn

________1 − r

n__2

1__2

c_____sin C

b_____sin B

a_____sin A

(h,k) = center, r = radius

axis of symmetry x = h

focus �h, k + �, directrix y = k −

axis of symmetry y = k

focus �h + , k�, directrix x = h −

foci (h ± c, k) where c 2 = a 2 − b 2

foci (h, k ± c) where c 2 = a 2 − b 2

foci (h ± c, k) where c 2 = a 2 + b 2

foci (h, k ± c) where c 2 = a 2 + b 2

an = nth terma1 = first termn = number of the termd = common differencer = common ratiosn = sum of the first n termss = sum of all the terms

1___4a

1___4a

1___4a

1___4a

continued

s = semi-perimeter = (a + b + c)_________2

b

aC

A

c

B

�Area = ����������������������������������s(s − a)(s − b)(s − c)

A = amount of money after t yearsp = starting principal r = interest rate

n = compound periods per yeart = number of years e ≈ 2.718

Nt = value after t time periodsr = rate of growth t = time periods

+ = 1, a > b

+ = 1, a > b

− = 1

− = 1

A = p �1 + �nt

A = pert

Nt = N0(1 + r)t, Nt = N0ert

r__n

(x − h)2________

b 2

(y − k)2________

a 2

(y − k)2________

b 2

(x − h)2________

a 2

(x − h)2________

b 2

(y − k)2________

a 2

(y − k)2________

b 2

(x − h)2________

a 2

Page 16: PreCalcs QC part 1 and Key

Polar Coordinates and Vectors

De Moivre’s Theorem [r(cos θ + i sin θ)]n = r n(cos nθ + i sin nθ)

Conversion: Polar to x = r cos θRectangular Coordinates y = r sin θ

Conversion: Rectangularto Polar Coordinates

Product of ComplexNumbers in Polar Form

Inner Product of Vectors

Matrices

Determinant of a 2×2 Matrix det� � = ad − bc

Determinant of a 3×3 Matrix det� � = a•det� � − b•det� � + c•det� �

Inverse of a 2×2 Matrix M −1 = � � where M = � �

Trigonometry

Sum and Difference Identities sin(α ± β ) = sin α cos β ± cos α sin βcos(α ± β ) = cos α cos β sin α sin β

tan(α ± β ) =

Double-Angle Identities sin 2θ = 2 sin θ cos θcos 2θ = cos2 θ − sin2 θ

tan 2θ =

Half-Angle Identities sin = ±

cos = ±

tan = ± , where cos α ≠ −11 − cos α_________1 + cos α

α__2

1 + cos α_________2

α__2

1 − cos α_________2

α__2

2 tan θ_________1 − tan2 θ

tan α ± tan β_____________1 ± tan α tan β

±bd

ac

−ca

d−b

1______det M

eh

dg

fj

dg

fj

eh

cfj

beh

adg

bd

ac

r = radius, distance from originθ = angle in standard positionn = exponent

a = ⟨a1,a2⟩ vector in the planea = ⟨a1,a2,a3⟩ vector in space

α, β, θ = angles, frompositive x-axis

r = , θ = arctan , when x > 0

θ = arctan + π, when x < 0

r1(cos θ1 + i sin θ1)•r2(cos θ2 + i sin θ2) =r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]

a•b = a1b1 + a2b2 + a3b3

y__x

y__x

��������x 2 + y 2

±

© 2010 by ACT, Inc. All rights reserved.14195 *0190D1080* Rev 2

Page 17: PreCalcs QC part 1 and Key

Answer Key

1) A2) C3) B4) D5) D6) B7) C8) C9) C

10) D11) D12) B13) B14) B15) D16) A17) A18) B19) C20) A

Page 18: PreCalcs QC part 1 and Key

Scoring Criteria:

A 4-point response may include, but is not limited to, the following points:

A. Correct domain and range: Domain = [–4,∞), Range = [0,∞)

Explanation of how the answers were found: The domain is all x values for which 4x + is defined. Since the index is even, the radicand cannot be negative to have a

real-number result. I found the value for x that results in a radicand of zero, which is –4. Any value of x greater than or equal to −4 will result in a nonnegative radicand, so the domain is [–4,∞). The range is all y values on the graph. Since A is always > 0 for

0A ≥ , the range is [0,∞).

B. Correct points: (–4,0), (0,2), (4,2.8), (8,3.5)

Note: There is an infinite number of correct points in addition to these given that students may find.

Correct graph:

Appropriate work needed to find the correct points:

f(–4) = 4 4 0 0− + = =

f(0) = 0 4 4 2+ = =

f(4) = 4 4 8 2.8+ = =

f(8) = 8 4 12 3.5+ = =

C. Explanation of how the graphs relate: The graph of f(x) = 4x + is the graph of g(x) = x shifted 4 units to the left.

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Page 19: PreCalcs QC part 1 and Key

Rubric: 4 A response at this level provides evidence of thorough knowledge and

understanding of the subject matter. • The response addresses all parts of the question or problem correctly. • The response demonstrates efficient and accurate use of appropriate procedures. • The explanation of strategies used in the response shows evidence of a good

understanding of mathematical concepts and principles, and it does not contain any misconceptions.

• The explanation in the response is clear and coherent. 3 A response at this level provides evidence of competent knowledge and

understanding of the subject matter. • The response addresses most parts of the question or problem correctly. • The response includes some minor errors but generally uses appropriate procedures

accurately. • The explanation of strategies used in the response shows some evidence of a good

understanding of mathematical concepts and principles, and it contains few, if any, misconceptions.

• The explanation in the response is mostly clear and coherent. 2 A response at this level provides evidence of a basic knowledge and

understanding of the subject matter. • The response addresses some parts of the question or problem correctly. • The response includes a number of errors but demonstrates some use of

appropriate procedures. • The explanation of strategies used in the response shows a little evidence of

understanding of mathematical concepts and principles, but it may contain some evidence of misconceptions.

• The explanation in the response is partially clear, but some parts may be difficult to understand.

1 A response at this level provides evidence of minimal knowledge and

understanding of the subject matter. • The response addresses a few parts of the problem correctly, but the response is

mostly incorrect. • The response includes inappropriate procedures or simple manipulations that show

little or no understanding of correct procedures. • The explanation of strategies used in the response shows little or no evidence of

understanding of mathematical concepts and principles, and it may contain evidence of significant misconceptions.

• Many parts of the explanation are difficult to understand. 0 A response at this level is not scorable. The response is off-topic, blank, hostile, or

otherwise not scorable.

Page 20: PreCalcs QC part 1 and Key

Scoring Criteria:

A 4-point response may include, but is not limited to, the following points:

A. Correct graph:

B. Correct answer: f(2) = 3

Appropriate work needed to find the answer:

f(2) = 12 (2) + 2

f(2) = 1 + 2

C. Correct explanation: A function has exactly 1 value in the range for each value in the domain. Because 2x + 1 and 1

2 x + 2 have constraints that include the values –1 and 2, x2 – 2 must exclude these values.

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Page 21: PreCalcs QC part 1 and Key

Rubric: 4 A response at this level provides evidence of thorough knowledge and

understanding of the subject matter. • The response addresses all parts of the question or problem correctly. • The response demonstrates efficient and accurate use of appropriate procedures. • The explanation of strategies used in the response shows evidence of a good

understanding of mathematical concepts and principles, and it does not contain any misconceptions.

• The explanation in the response is clear and coherent. 3 A response at this level provides evidence of competent knowledge and

understanding of the subject matter. • The response addresses most parts of the question or problem correctly. • The response includes some minor errors but generally uses appropriate procedures

accurately. • The explanation of strategies used in the response shows some evidence of a good

understanding of mathematical concepts and principles, and it contains few, if any, misconceptions.

• The explanation in the response is mostly clear and coherent. 2 A response at this level provides evidence of a basic knowledge and

understanding of the subject matter. • The response addresses some parts of the question or problem correctly. • The response includes a number of errors but demonstrates some use of

appropriate procedures. • The explanation of strategies used in the response shows a little evidence of

understanding of mathematical concepts and principles, but it may contain some evidence of misconceptions.

• The explanation in the response is partially clear, but some parts may be difficult to understand.

1 A response at this level provides evidence of minimal knowledge and

understanding of the subject matter. • The response addresses a few parts of the problem correctly, but the response is

mostly incorrect. • The response includes inappropriate procedures or simple manipulations that show

little or no understanding of correct procedures. • The explanation of strategies used in the response shows little or no evidence of

understanding of mathematical concepts and principles, and it may contain evidence of significant misconceptions.

• Many parts of the explanation are difficult to understand. 0 A response at this level is not scorable. The response is off-topic, blank, hostile, or

otherwise not scorable.