College Algebra Final Exam Review: Comprehensive Study Notes

Equations & Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable using algebraic operations.

• Key Point 1: Combine like terms and use inverse operations to isolate the variable.

• Key Point 2: Check your solution by substituting it back into the original equation.

• Example: Solve $-2(x+4)-2x=9(x/2)-4$.

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Key Point 1: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

• Key Point 2: Factoring is possible when the equation can be written as a product of two binomials.

• Example: Solve $x^2 - 10x = 3$.

Solving Inequalities

Inequalities involve finding the set of values for which an expression is greater than, less than, or equal to another. Solutions are often expressed in interval notation and graphed on a number line.

• Key Point 1: Use algebraic manipulation similar to equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Key Point 2: Express solutions in interval notation, e.g., $(-\infty, 2]$.

• Example: Solve $20-2x \geq 4(x+2)-4x$.

Functions

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

• Key Point 1: For rational functions, exclude values that make the denominator zero.

• Key Point 2: Use interval notation to express domain and range.

• Example: For $f(x) = \frac{x-1}{x^2-3}$, the domain is all real numbers except $x = \pm\sqrt{3}$.

Intercepts and Asymptotes

Intercepts are points where the graph crosses the axes. Asymptotes are lines that the graph approaches but never touches.

• Key Point 1: The y-intercept is found by setting $x=0$.

• Key Point 2: Vertical asymptotes occur where the denominator of a rational function is zero.

• Example: For $f(x) = \frac{x-1}{x^2-3}$, vertical asymptotes at $x = \pm\sqrt{3}$.

Function Operations and Inverses

Functions can be added, subtracted, multiplied, divided, and composed. The inverse of a function reverses the roles of input and output.

• Key Point 1: The inverse function $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$.

• Key Point 2: To find the inverse, solve $y = f(x)$ for $x$ and interchange $x$ and $y$.

• Example: Find the inverse of $f(x) = 4x - 8$.

Polynomial and Rational Functions

End Behavior and Zeros

The end behavior of a polynomial describes how the function behaves as $x$ approaches infinity or negative infinity. The zeros are the x-values where the function equals zero.

• Key Point 1: The degree and leading coefficient determine end behavior.

• Key Point 2: The maximum number of real zeros is equal to the degree of the polynomial.

• Example: For $f(x) = x^3 - 3x^2 - 2x^2 - 3$, degree is 3, so up to 3 real zeros.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form $(x - c)$.

• Key Point 1: Write coefficients, use the zero of the divisor, and perform the synthetic division steps.

• Key Point 2: The remainder indicates whether the divisor is a factor.

• Example: Divide $5x^3 - 6x^2 - 28x + 8$ by $x + 2$.

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Solutions require finding a common denominator and checking for extraneous solutions.

• Key Point 1: Multiply both sides by the least common denominator to clear fractions.

• Key Point 2: Check solutions to ensure they do not make any denominator zero.

• Example: Solve $\frac{x}{x^2-2x-3} = \frac{2x-5}{x^2-2x-3}$.

Exponential & Logarithmic Functions

Exponential Equations

Exponential equations have variables in the exponent. They are solved using logarithms.

• Key Point 1: Take the logarithm of both sides to bring down the exponent.

• Key Point 2: Use properties of logarithms to simplify.

• Example: Solve $0.8^x = 4$.

Logarithmic Equations

Logarithmic equations involve the logarithm of a variable. Use properties of logarithms to combine and solve for the variable.

• Key Point 1: Use the product, quotient, and power rules for logarithms.

• Key Point 2: Convert to exponential form to solve.

• Example: Solve $\log_2(x) + \log_2(x) = 2$.

Compound Interest

Compound interest problems use exponential functions to model the growth of investments over time.

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

• Key Point 2: For continuous compounding, use $A = Pe^{rt}$.

• Example: $25,000$ invested at $2.56\%$ compounded quarterly for $25$ years.

Systems of Equations & Matrices

Solving Systems of Equations

Systems of equations are sets of equations with multiple variables. Solutions can be found using substitution, elimination, or matrices.

• Key Point 1: Substitution involves solving one equation for a variable and substituting into the other.

• Key Point 2: Elimination involves adding or subtracting equations to eliminate a variable.

• Example: Solve $-3x + 2y = 11$ and $x - y = 9$.

Graphs of Equations & Functions

Scatter Diagrams and Function Models

Scatter diagrams display data points and help identify the type of function that best models the data (linear, quadratic, exponential, etc.).

• Key Point 1: Linear models show a straight-line pattern.

• Key Point 2: Quadratic models show a parabolic pattern.

• Example: Match the scatter diagram to the function type: cubic, exponential, linear, logarithmic, quadratic.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections of function graphs.

• Key Point 1: $f(x-h)$ shifts the graph right by $h$ units; $f(x)+k$ shifts up by $k$ units.

• Key Point 2: $-f(x)$ reflects the graph over the x-axis.

• Example: State the transformations to go from $f(x)$ to $f(x-2)-3$.

Applications: Cost, Revenue, and Optimization

Cost and Revenue Functions

Cost and revenue functions model business scenarios, such as break-even analysis and profit maximization.

• Key Point 1: Cost function: $C(x) = \text{fixed cost} + \text{variable cost} \times x$.

• Key Point 2: Revenue function: $R(x) = \text{price per unit} \times x$.

• Example: If fixed cost is $150,000$ and variable cost is $18$per unit,$C(x) = 150,000 + 18x$.

Optimization Problems

Optimization involves finding maximum or minimum values, often using quadratic or other functions.

• Key Point 1: Express the quantity to be maximized/minimized as a function.

• Key Point 2: Use calculus or vertex formula for quadratics to find the optimal value.

• Example: A farmer with $1000$ft of fencing wants to maximize area:$A = x(1000-2x)$.

Formulas Reference

Formula	Description
$y = mx + b$	Slope-intercept form of a line
$y - y_1 = m(x - x_1)$	Point-slope form of a line
$f(x) = a(x-h)^2 + k$	Vertex form of a quadratic
$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$	Quadratic formula
$A = P(1 + \frac{r}{n})^{nt}$	Compound interest
$A = Pe^{rt}$	Continuous compounding
$y = \log_b x$	Logarithmic function
$\log_a x = b \iff x = a^b$	Logarithm-exponential equivalence
$\log_a(xy) = \log_a x + \log_a y$	Product rule for logarithms
$\log_a(\frac{x}{y}) = \log_a x - \log_a y$	Quotient rule for logarithms
$\log_a(x^r) = r \log_a x$	Power rule for logarithms