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Questions and Answers
What is the Compounded Interest Formula?
PERT, where P = Starting Amount, R = Interest Rate, T = Time (Years)
What is the Interest Formula?
P(1+r/n)^(nt), where P = Starting Amount, R = Interest Rate, N = Number of Times Per Year, T = Time (Years)
What is the Change of Base Formula?
logbA = logA/logB
What is the definition of a logarithm?
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What does it mean to exponentiate and reverse?
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What is the formula for Direct Variation?
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What is the formula for Inverse Variation?
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What is the formula for Join Variation?
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What are the Horizontal Asymptote Rules for Rational Functions?
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What are the steps to find the Distance Between Parallel Lines?
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What is the Circle Formula?
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What is the Parabola Formula?
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What is the formula for an Ellipse?
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What is the formula for a Hyperbola?
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What are the formulas for Arithmetic sequences?
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What are the formulas for Geometric sequences?
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What is the Law of Sines?
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What is the Law of Cosines?
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How many solutions can sine have?
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What is the equation for a Sine Graph?
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What is the equation for a Cosine Graph?
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What is the equation for a Tangent Graph?
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Study Notes
Compounded Interest
- Formula: ( PERT ) where ( P ) is the starting amount, ( R ) is the interest rate, and ( T ) represents time in years.
Interest Calculation
- Formula: ( P(1+\frac{r}{n})^{nt} )
- Variables: ( P ) (starting amount), ( r ) (interest rate), ( n ) (payments per year), and ( t ) (time in years).
Logarithmic Functions
- Change of Base Formula: ( \log_b A = \frac{\log A}{\log B} )
- Definition of a Logarithm: ( \log_b A = C ) implies ( b^C = A ) where ( b ) is the base, ( A ) the number, and ( C ) the exponent.
Exponential and Logarithmic Relationships
- Exponentiation involves expressing terms as powers to eliminate logarithms.
- Reversing the process puts terms back into logarithmic form to isolate exponents.
Variation Concepts
- Direct Variation: ( y = kx ) where ( k ) is the constant of variation.
- Inverse Variation: ( y = \frac{k}{x} ).
- Joint Variation: ( z = kxy ) indicating dependency on multiple variables.
Analyzing Rational Functions
- Asymptote Rules:
- ( n > m ): Slant asymptote, no horizontal asymptote.
- ( n = m ): Horizontal asymptote at ( y = \frac{n}{m} ).
- ( n < m ): Horizontal asymptote at ( y = 0 ).
Finding Distances
- Distance Between Parallel Lines:
- Find a perpendicular line and its intersection points on each line.
- Use the distance formula to calculate the distance between these points.
Conics Formulas
- Circle: ( (x-h)^2 + (y-k)^2 = r^2 ) where ( (h,k) ) is the center and ( r ) the radius.
- Parabola:
- Vertical: ( y = \frac{1}{4p}(x-h)^2 + k )
- Horizontal: ( x = \frac{1}{4p}(y-k)^2 + h )
- Parameter ( p ) represents the focal distance.
- Ellipse:
- Foci: ( c^2 = a^2 - b^2 )
- Horizontal form: ( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 )
- Vertical form: ( \frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1 )
- Hyperbola:
- Foci: ( c^2 = a^2 + b^2 )
- Horizontal form: ( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 )
- Vertical form: ( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 )
Sequences and Series
- Arithmetic Sequence:
- Recursive: ( a_n = a_{n-1} + d )
- Explicit: ( a_n = a_1 + (n-1)d )
- Summation: ( S_n = \frac{n}{2}(a_1 + a_n) )
- Geometric Sequence:
- Recursive: ( a_n = r \cdot a_{n-1} )
- Explicit: ( a_n = a_1 \cdot r^{n-1} )
- Finite Sum: ( S_n = \frac{a_1(1 - r^n)}{1 - r} ), Infinite Sum: ( S = \frac{a_1}{1 - r} )
Trigonometric Laws
-
Law of Sines: ( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} )
-
Area using sine: ( \text{Area} = \frac{1}{2}bc \sin A )
-
Law of Cosines:
- ( a^2 = b^2 + c^2 - 2bc \cos A )
- Similar formulas for ( b ) and ( c ).
Solutions in Triangle Geometry
- Number of Solutions (Sine):
- For obtuse angle ( A ): one solution.
- For acute angle ( A ): conditions on height ( h ) determine the number of solutions.
Graphing Trigonometric Functions
- Sine Graph: ( y = a \sin b(x-h) + k ) with amplitude ( a ) and period ( \frac{2\pi}{b} ), starts at 0.
- Cosine Graph: ( y = a \cos b(x-h) + k ), starts at the highest or lowest point based on reflection.
- Tangent Graph: ( y = a \tan b(x-h) + k ), has a period of ( \frac{\pi}{2} ) and numerous asymptotes.
Studying That Suits You
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Description
Test your understanding of key concepts in Algebra 2 with these flashcards. Covering topics such as the compounded interest formula and logarithmic functions, these cards are designed to help reinforce your knowledge for the upcoming exam.