College Algebra Practice Test 3: Rational Functions, Exponentials, Logarithms, and Systems of Equations

Rational Functions and Asymptotes

Identifying Asymptotes of Rational Functions

Rational functions are quotients of polynomials. Their graphs often have vertical, horizontal, or oblique asymptotes, which describe the behavior of the function as the input approaches certain values.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes: Determined by comparing the degrees of the numerator and denominator.

• Oblique (Slant) Asymptotes: Occur when the degree of the numerator is exactly one more than the degree of the denominator.

Example: For $f(x) = \frac{2x-6}{x^2-x-12}$, set $x^2-x-12=0$ to find vertical asymptotes.

Graphing Functions Using Transformations

Transformations of Exponential and Logarithmic Functions

Transformations shift, stretch, or reflect the basic graphs of functions. For exponential functions, vertical shifts and reflections are common.

• Vertical Shift: $f(x) = a^{x} + k$ shifts the graph up by $k$ units.

• Reflection: $f(x) = -a^{x}$ reflects the graph across the x-axis.

Example: $f(x) = -3^{x} + 2$ is a reflection of $3^{x}$ across the x-axis, then shifted up by 2 units.

Exponential Growth and Compound Interest

Compound Interest Formulas

Compound interest describes how money grows when interest is added to the principal periodically.

• Periodic Compounding: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

• Doubling Time (Continuous): $t = \frac{\ln 2}{r}$

Example: $P = 10,000$, $r = 0.08$, $t = 8$ years. Calculate $A$ for quarterly, monthly, and continuous compounding.

Logarithmic Equations and Properties

Solving Logarithmic Equations

Logarithmic equations can be solved by rewriting them in exponential form or using properties of logarithms.

• Change of Base: $\log_b a = \frac{\ln a}{\ln b}$

• Product Rule: $\log_b(xy) = \log_b x + \log_b y$

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

• Power Rule: $\log_b(x^k) = k \log_b x$

Example: Solve $\log_2 x = 3$ by rewriting as $x = 2^3 = 8$.

Evaluating Logarithms

Exact and Approximate Values

Some logarithms can be evaluated exactly, while others require a calculator.

• Exact Values: $\log_{10} 0.0001 = -4$ because $10^{-4} = 0.0001$.

• Approximate Values: Use a calculator for values like $\log 56$ or $\ln 2$.

Graphing Logarithmic Functions

Transformations of Logarithmic Graphs

Logarithmic functions can be shifted horizontally and vertically.

• Horizontal Shift: $g(x) = \log_b(x + h)$ shifts the graph left by $h$ units.

• Vertical Shift: $g(x) = \log_b x + k$ shifts the graph up by $k$ units.

Example: $g(x) = \log_2(x + 3)$ is the graph of $\log_2 x$ shifted left by 3 units.

Solving Exponential Equations

Equating Exponents and Bases

Exponential equations can often be solved by expressing both sides with the same base and equating exponents.

• General Form: If $a^{f(x)} = a^{g(x)}$, then $f(x) = g(x)$.

• Using Logarithms: If bases differ, take logarithms of both sides.

Example: Solve $3^{x+1} = 9^{2x}$ by rewriting $9$as $3^2$.

Systems of Equations

Solving Linear Systems

Systems of equations can be solved by substitution, elimination, or using matrices.

• Substitution: Solve one equation for a variable and substitute into others.

• Elimination: Add or subtract equations to eliminate a variable.

• Matrix Methods: Write the system as an augmented matrix and use Gaussian or Gauss-Jordan elimination.

Example: Solve the system: $\begin{cases} x + y + z = 3 \\ 2x + y + z = 6 \\ x + 2y + 3z = 2 \end{cases}$

Matrices: Operations, Inverses, and Determinants

Matrix Arithmetic and Properties

Matrices are rectangular arrays of numbers used to represent systems of equations and perform linear transformations.

• Addition/Subtraction: Add or subtract corresponding elements.

• Multiplication: Multiply rows by columns.

• Inverse: $A^{-1}$ exists if $A$ is square and $\det(A) \neq 0$.

• Determinant: For a $3 \times 3$ matrix, $\det(A)$ can be found using cofactor expansion.

Example: Find the inverse of $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & -1 & -1 \end{bmatrix}$ and the determinant of $\begin{bmatrix} 1 & 0 & 0 & -2 \\ 0 & 1 & 2 & 1 \\ 0 & -1 & -1 & 0 \end{bmatrix}$.

Summary Table: Key Formulas

Topic	Formula	Description
Compound Interest (Periodic)	$A = P\left(1 + \frac{r}{n}\right)^{nt}$	Amount after $t$ years, $n$ compounding periods/year
Compound Interest (Continuous)	$A = Pe^{rt}$	Amount after $t$ years, continuous compounding
Doubling Time (Continuous)	$t = \frac{\ln 2}{r}$	Time to double investment
Logarithm Product Rule	$\log_b(xy) = \log_b x + \log_b y$	Logarithm of a product
Logarithm Quotient Rule	$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$	Logarithm of a quotient
Logarithm Power Rule	$\log_b(x^k) = k \log_b x$	Logarithm of a power
Matrix Inverse	$A^{-1}$	Inverse exists if $\det(A) \neq 0$
Determinant (3x3)	Use cofactor expansion	Determines invertibility

Additional info: These topics correspond to College Algebra chapters on rational functions, exponential and logarithmic functions, systems of equations, and matrices. The questions cover both conceptual understanding and computational skills.