Algebra IIA Study Notes

GP3 Test: Algebra IIA 2 Study Notes

Rational Exponents and Radical Functions (13 Questions)

• Evaluating a Rational Exponent

  • A rational exponent is of the form $a^{\frac{m}{n}}$, where:

  • $a$ is the base.

  • $m$ is the numerator (an integer).

  • $n$ is the denominator (a positive integer).

  • It can be rewritten using radicals:

  • $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

• Solving an Equation with a Rational Exponent

  • Approach: To solve an equation involving a rational exponent, isolate the term with the exponent and raise both sides to the reciprocal of the exponent.

• Simplifying Radicals

  • Multiplying 2 Radicals

  • Rule: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.

  • Example: $\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6$.

  • Single Radical with Variables

  • Example Simplification:

    • $\sqrt{x^2} = x$, where $x \geq 0$.

• Radical Operations

  • Adding & Subtracting

  • Like terms only: $\sqrt{a} + \sqrt{a} = 2\sqrt{a}$;

  • Example: $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$.

  • Multiplying Binomials

  • Use the distributive property or FOIL:

    • Example: $(\sqrt{a} + \sqrt{b})(\sqrt{c} + \sqrt{d})$.

  • Dividing by a Binomial

  • Rationalize the denominator: $\frac{1}{\sqrt{a} + b}$.

• Graphing a Square Root Function

  • General form: $f(x) = \sqrt{x}$.

  • Domain: $x \geq 0$; Range: $y \geq 0$.

  • Plot key points: (0,0), (1,1), (4,2), (9,3).

• Radical Equation

  • An equation that contains a variable within a radical.

  • Example: $\sqrt{x+3} = 5$.

  • To solve, square both sides to remove the radical: $x + 3 = 25$.

• Function Operations

  • Subtracting Functions

  • $f(x) - g(x)$; ensure proper domain for each function.

  • Multiplying Functions

  • $f(x) \cdot g(x)$; example: $f(x) = 2x, g(x) = x^2 \Rightarrow f(x) \cdot g(x) = 2x^3$.

• Function Composition Word Problem

  • Describes a situation involving nested functions.

  • Example: If $f(x) = 2x$ and $g(x) = x^2$, then $f(g(3)) = f(9) = 18$.

• Finding & Graphing an Inverse Function

  • The inverse of function $f$ is denoted as $f^{-1}$.

  • To find the inverse, switch $x$ and $y$, then solve for $y$.

  • Graphically, the inverse is a reflection over the line $y=x$.

Exponential and Logarithmic Functions (12 Questions)

• Graphing an Exponential Growth Function

  • General form: $f(x) = a \cdot b^x$ (where b > 1).

  • Key characteristics:

  • Y-intercept at (0, a).

  • Asymptote at y = 0.

• Compound Interest Formula

  • The formula for compound interest is given by:
    $A = P \left(1 + \frac{r}{n}\right)^{nt}$
    where:

  • $A$ = the amount of money accumulated after n years, including interest.

  • $P$ = principal amount (the initial amount of money).

  • $r$ = annual interest rate (decimal).

  • $n$ = number of times that interest is compounded per year.

  • $t$ = time the money is invested for in years.

• Exponential Equations

  • Solutions to equations of the form $b^x = k$ involve taking logarithms:

  • Example: Solve for $x$ in equation $2^x = 16$ by converting to logarithmic form:

    • $x = \log_{2}(16) = 4$.

• Common Base

  • Sometimes equations can be expressed with a common base for simplification.

• Convert to a Logarithm

  • Use conversion: $b^x = k$ translates to $x = \log_{b}(k).$

• Base e

  • When the base of logarithm is $e$ (Euler's number), refer to it as a natural logarithm:

  • Denoted as $\ln(x)$; for example: $\ln(e) = 1$.

• Graphing a Logarithmic Function

  • General from: $f(x) = \log_{b}(x)$.

  • Characteristics:

  • Domain: x > 0.

  • Range: All real numbers.

  • Asymptote at $x = 0$.

• Logarithmic Equations

  • 1 Logarithm: Solve equations with a single logarithm.

  • Logarithm on Both Sides: For example: $\log<em>{b}(x) = \log</em>{b}(y)$ implies $x = y$.

  • Use Properties of Logarithms:

  • Power & Quotient Property:

    • Power: $\log<em>{b}(x^n) = n\log</em>{b}(x)$.

    • Quotient: $\log<em>{b}(\frac{x}{y}) = \log</em>{b}(x) - \log_{b}(y)$.

  • Product Property:

    • $\log<em>{b}(xy) = \log</em>{b}(x) + \log_{b}(y)$.

  • Natural Logarithm:

  • Special case of logarithm where the base is $e$.

• Continuous Compound Interest Formula

  • The formula for continuous compounding is:
    $A = Pe^{rt}$
    where:

  • $A$ = the future value of the investment/loan, including interest.

  • $P$ = the principal investment amount (initial deposit or loan amount).

  • $r$ = the annual interest rate (decimal).

  • $t$ = the time in years the money is invested or borrowed for.

  • Example calculation:

  • If $P = 1000$, $r = 0.05$, and $t = 5$, then

  • $A = 1000e^{(0.05)(5)} = 1000e^{0.25}$ approximately equals 1000×1.284 = $1284.