Algebra 2 Exam Flashcards

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.