College Algebra Final Exam Review: Comprehensive Study Notes

Graphs, Functions, and Models

Understanding and Graphing Functions

Functions are mathematical relationships where each input has exactly one output. Graphing functions helps visualize their behavior and key features.

• Definition of a Function: A relation where each input (x-value) has one and only one output (y-value).

• Graphing: Plot points for various x-values and connect them smoothly to reveal the function's shape.

• Example: For $f(x) = 2x - 3$, plot points for several x-values and draw the line.

Symmetry of Graphs

Determining symmetry helps classify functions and predict their behavior.

• Symmetry about the y-axis: If $f(-x) = f(x)$ for all x, the graph is symmetric about the y-axis (even function).

• Symmetry about the x-axis: If replacing y with -y yields the same equation, the graph is symmetric about the x-axis (not a function).

• Symmetry about the origin: If $f(-x) = -f(x)$, the graph is symmetric about the origin (odd function).

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Definition: A function defined by multiple sub-functions, each applying to a certain interval.

• Example: $f(x) = \begin{cases} x^2 & x < -2 \\ 2 & -2 \leq x \leq 2 \\ x+2 & x > 2 \end{cases}$

• Graph each piece over its specified interval.

More on Functions

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a function.

• Vertical and Horizontal Shifts: $f(x) + k$ shifts up/down; $f(x - h)$ shifts right/left.

• Reflections: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Stretches/Compressions: $af(x)$ stretches (|a| > 1) or compresses (0 < |a| < 1) vertically.

• Example: The graph of $y = \sqrt{x-3} - 2$ is the graph of $y = \sqrt{x}$ shifted right 3 units and down 2 units.

Domain and Range

The domain is the set of all possible input values; the range is the set of all possible output values.

• Finding Domain: Exclude values that make the function undefined (e.g., division by zero, negative under even roots).

• Finding Range: Analyze the outputs based on the domain and the function's behavior.

Function Operations and Composition

Functions can be added, subtracted, multiplied, divided, or composed to create new functions.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

Quadratic Functions and Equations; Inequalities

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex: The highest or lowest point, at $x = -\frac{b}{2a}$

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$

• Intercepts: Set $x=0$ for y-intercept; set $f(x)=0$ for x-intercepts.

• Example: For $f(x) = x^2 - 4x + 3$, vertex at $x = 2$, y-intercept at $f(0) = 3$.

Solving Quadratic Equations

• Factoring: Express as $(x - r_1)(x - r_2) = 0$ and solve for $x$.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Completing the Square: Rewrite in the form $(x-h)^2 = k$ and solve for $x$.

Inequalities

• Solving Linear and Quadratic Inequalities: Find critical points, test intervals, and write the solution in interval notation.

• Example: Solve $x^2 - 5x + 6 > 0$ by finding roots and testing intervals.

Polynomial and Rational Functions

Polynomial Functions

Polynomials are sums of terms with non-negative integer exponents. Their graphs can have multiple turning points.

• End Behavior: Determined by the leading term $a_nx^n$.

• Zeros: Values of $x$ where $f(x) = 0$.

• Factoring: Express as a product of lower-degree polynomials.

Rational Functions

Rational functions are ratios of polynomials. They may have vertical, horizontal, or oblique asymptotes.

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

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

• Holes: Occur where factors cancel in numerator and denominator.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = ab^{x}$, where $b > 0$, $b \neq 1$.

• Growth and Decay: If $b > 1$, the function grows; if $0 < b < 1$, it decays.

• Applications: Population growth, radioactive decay, compound interest.

• Example: $P(t) = P_0e^{kt}$ models continuous growth or decay.

Logarithmic Functions

Logarithms are the inverses of exponentials: $y = \log_b x$ means $b^y = x$.

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

• Change of Base: $\log_b x = \frac{\log_a x}{\log_a b}$

Systems of Equations and Matrices

Solving Systems of Equations

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

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

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

• Matrices: Write the system as $AX = B$ and solve using matrix operations.

Conic Sections

Circles

The equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

• Given a point on the circle and the center, use the distance formula to find $r$.

Sequences, Series, and Combinatorics

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers following a pattern.

• Arithmetic Sequence: Each term differs by a constant $d$; $a_n = a_1 + (n-1)d$

• Geometric Sequence: Each term is multiplied by a constant $r$; $a_n = a_1 r^{n-1}$

Series

• Sum of Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

Applications and Problem Solving

Word Problems and Modeling

Many real-world problems can be modeled using algebraic equations and functions.

• Cost Functions: $C(x) = \text{fixed cost} + \text{variable cost} \times x$

• Coin Problems: Set up equations based on the value and number of coins.

• Projectile Motion: $h(t) = -16t^2 + v_0 t + h_0$ models the height of an object under gravity.

Graph Analysis

Analyzing graphs involves determining domain, range, intervals of increase/decrease, and identifying maxima/minima.

• Increasing/Decreasing: Where the graph rises or falls as x increases.

• Relative Extrema: Local maximum or minimum points.

Summary Table: Key Algebraic Concepts

Concept	Definition	Key Formula	Example
Quadratic Formula	Solves $ax^2 + bx + c = 0$	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 4x + 3 = 0$
Circle Equation	All points $(x, y)$ at distance $r$ from $(h, k)$	$(x-h)^2 + (y-k)^2 = r^2$	Center (2, 3), $r = 5$
Arithmetic Sequence	Constant difference $d$	$a_n = a_1 + (n-1)d$	$2, 5, 8, 11, ...$
Exponential Growth	Constant percent increase	$P(t) = P_0e^{kt}$	$P_0 = 1000, k = 0.05$

Additional info:

• These notes are based on a comprehensive final exam review covering all major College Algebra topics, including graphing, transformations, solving equations and inequalities, systems, conic sections, sequences, and applications.

• Students should practice graphing, solving, and interpreting functions, as well as applying algebraic concepts to real-world scenarios.