Pre-Calculus Algebra Final Exam Review

Continuous Compound Interest

• Use the formula A = Pe^(rt) to calculate the time for an investment to grow.
• For $900 to grow to $14,700 at a 5.7% interest rate, solve for t using continuous compounding.

Exponential Growth Model

• Population growth can be modeled with p(t) = 10,472e^(0.004t).
• To determine when the population reaches 15,708, isolate t and solve.

Parabola Equations

• Identify the equation of a parabola by examining shifts in the graph.
• The form y = (x + h)² + k indicates horizontal shift by h and vertical shift by k.

Logarithmic Properties

• The expression 3log(4x) can be rewritten by raising the inside to the power of 3, yielding log(64x³).
• Understanding properties of logarithms is essential for algebraic manipulation.

Exponential Growth Calculation

• For the population modeled by p(t) = 9779e^(0.004t), calculate population after 10 years by substituting t = 10.

Cost Optimization

• The cost function c(x) = 4x² - 312x + 49 can be analyzed to find the minimum cost by calculating vertex form using y = -b/(2a).

Radioactive Decay

• The decay of a substance is modeled by A(t) = 500e^(-0.153t).
• Determine the time for the sample to decay to half its original amount using the decay function.

Interest Rate Calculation

• For compounding interest, use r = n((A/P)^(1/nt) - 1) to find the interest rate needed for investments to grow.

Population Growth Rate

• The formula N = Ie^(kt) models population growth, where k represents the growth constant.
• Solve for the time required to increase from 69 million to 95 million cars, given a 4.1% annual growth rate.

Area and Perimeter of Rectangles

• The area of a rectangular plot is given as 126 yd² with a perimeter of 46 yd.
• Solve for side lengths by testing values within the established constraints.

Investment Doubling Time

• To find the rate when $3171 doubles in 14 years, apply the quarterly compounding interest formula.

Quadratic Equations

• Simplify the quadratic equation to the standard form ax² + bx + c.
• Use the quadratic formula x = (-b ± √(b² - 4ac))/(2a) to find solutions.

Logarithmic Conversion

• Convert exponential expressions to logarithmic form, using the identity a^b = c corresponds to log_a(c) = b.

Solving Radical Equations

• To solve √(x + 3) = x - 3, square both sides and apply quadratic solving methods.

Rational Equation Solutions

• For equations like (9x/(x - 9)) - (4/x) = 36/(x² - 9x), combine and simplify fractions then factor.

Absolute Value Equations

• For equations involving absolute values, isolate and simplify to determine possible values of x.

Exponential Equations

• Solve equations of the form e^(x - 2) = (1/e^(6))^(x + 3) by applying properties of exponents for simplification.