Advanced Algebra Final Exam Flashcards

Parabolas

• Vertex is calculated using the formula -b/2a.
• X-intercepts are found by setting y=0; solve for x.
• Y-intercepts are found by setting x=0; solve for y.
• Axis of symmetry is the k value from the vertex notation (vertex = (h, k)).
• Standard equation for a parabola is y = a(x-h)² + k.
• To find the maximum point of a parabola, use a graphing calculator to locate the maximum value on the equation.

Understanding Fractional Equations

• For equations resembling fractions, the domain includes all possible x values.
• X-intercepts are found by setting the numerator to 0, while Y-intercepts are found by making x=0.
• Asymptotes occur when the denominator equals 0.

Logarithmic Functions

• If no base is indicated in a logarithm, it is log base 10 (log10).
• When expanding logarithms involving an exponent, the exponent is brought in front: for example, y³ = 3 log y.
• In condensing logarithmic expressions, negative values move to the denominator.
• Fractions in front of logarithmic functions imply manipulation with radicals: the numerator goes inside the radical, and the denominator goes outside.

Logarithmic Properties

• Expanding log(xy) results in log x + log y.
• The addition of logarithms corresponds to multiplication, while subtraction indicates division.
• Steps to solve a logarithm for x:
  • First: isolate by adding or subtracting the extra value.
  • Second: divide out the extra value.
  • Third: convert to the exponential form and solve using a calculator.

Transitions Between Logarithmic Forms

• When converting values into logs:
  • Place log first.
  • The initial number becomes the new exponent.
  • The previous answer becomes the base.
  • The previous exponent converts to the new answer.
• For conversion from logarithmic form to exponential:
  • The original exponent becomes the base of the number.
  • The previous answer converts to the exponent.
  • The base number becomes the new resulting number.

Compound Interest Formula

• The formula for compound interest is A = P(1 + r/n)^(nt).
• Variables in this formula represent:
  • A: the amount of money accumulated after n years, including interest.
  • P: the principal amount (initial investment).
  • r: annual interest rate (decimal).
  • n: number of times that interest is compounded per year.
  • t: time the money is invested for, in years.

Domain and Range

• The domain represents the range of x values (from left to right).
• The range illustrates the vertical extent (up and down) and can be analyzed with a calculator.

Intercepts and Asymptotes

• Y-intercepts occur when x=0, and can be found using a calculator.
• X-intercepts occur when y=0, also found via calculator.
• The equation of vertical asymptotes is defined as y= the dotted line in the graph.

Miscellaneous

• The notation A = Pert indicates:
  • A: final amount.
  • P: initial principal.
  • e: base of the natural logarithm.
  • r: interest rate.
  • t: time in years.