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.

• Graphing Basic Functions: Plot points for given values of x and connect them smoothly to reveal the function's shape.

• Symmetry: A function may be symmetric with respect to the y-axis (even function), x-axis, or origin (odd function). Test symmetry by substituting -x for x or -y for y in the equation.

• Piecewise Functions: Defined by different expressions over different intervals. Graph each piece over its specified domain.

• Transformations: Shifting, reflecting, stretching, or compressing the graph of a function. For example, $f(x) + c$ shifts up, $f(x - h)$ shifts right.

Example: Graph $f(x) = x^2$ and $f(x) = (x-3)^2 - 5$ to see a right shift by 3 and a downward shift by 5.

More on Functions

Function Operations and Composition

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Composition: $(f \circ g)(x) = f(g(x))$ means substitute g(x) into f(x).

• Inverse Functions: If $f(g(x)) = x$ and $g(f(x)) = x$, then f and g are inverses. To find the inverse, solve for x in terms of y and interchange variables.

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$.

Quadratic Functions and Equations; Inequalities

Quadratic Functions and Their Properties

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

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

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

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

• Solving Quadratics: Use factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Inequalities: Solve quadratic inequalities by finding zeros and testing intervals.

Example: Solve $x^2 - 4x + 3 = 0$ by factoring: $(x-1)(x-3)=0$ so $x=1,3$.

Polynomial Functions and Rational Functions

Polynomial and Rational Equations

Polynomials are sums of terms with non-negative integer exponents. Rational functions are ratios of polynomials.

• End Behavior: Determined by the leading term for large $|x|$.

• Asymptotes: Vertical asymptotes occur where the denominator is zero; horizontal or oblique asymptotes depend on degrees of numerator and denominator.

• Solving Rational Equations: Multiply both sides by the least common denominator (LCD) to clear fractions.

Example: Solve $\frac{2}{x-1} = 3$ by multiplying both sides by $(x-1)$.

Exponential Functions and Logarithmic Functions

Exponential Growth and Decay

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

• Exponential Growth: $P(t) = P_0 e^{kt}$, where $k > 0$.

• Exponential Decay: $P(t) = P_0 e^{-kt}$, where $k > 0$.

• Logarithms: $y = \log_b x$ means $b^y = x$.

• Properties: $\log_b(xy) = \log_b x + \log_b y$, $\log_b(x^k) = k \log_b x$.

Example: Solve $2^x = 8$ by taking $\log_2$ of both sides: $x = 3$.

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 inverse matrices or row reduction.

Example: Solve $2x + y = 5$, $x - y = 1$ by elimination: Add equations to get $3x = 6$, so $x=2$, $y=1$.

Conic Sections

Circles, Parabolas, and More

Conic sections include circles, ellipses, parabolas, and hyperbolas, each with standard equations.

• Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: $y = a(x-h)^2 + k$

• Vertex: The center for circles, the turning point for parabolas.

Example: The circle with center (0,0) and radius 5: $x^2 + y^2 = 25$.

Sequences, Series, and Combinatorics

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers. Series are sums of sequences.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Summation: $S_n = \frac{n}{2}(a_1 + a_n)$ for arithmetic, $S_n = a_1 \frac{1 - r^n}{1 - r}$ for geometric ($r \neq 1$)

Example: Find the 5th term of $2, 4, 6, ...$: $a_5 = 2 + 4 \times 1 = 6$.

Application Problems

Word Problems and Modeling

Many algebraic concepts are applied to real-world problems, such as cost modeling, population growth, and coin problems.

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

• Population Growth: $P(t) = P_0 e^{kt}$

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

Example: If the setup fee is $225 and each shirt costs $6.75, $C(x) = 225 + 6.75x$.

Tables: Exponential Growth Example

Population Growth Table

The following table shows population growth over time, useful for modeling with exponential functions.

Year	Population
0	7,500
1	7,650
2	7,803
3	7,959
4	8,118
5	8,281
6	8,446
7	8,615

Additional info: Use the exponential growth formula $P(t) = P_0 e^{kt}$ to model the data.

Key Concepts and Strategies

• Always identify the type of function before solving or graphing.

• Check domains and ranges for all functions.

• Label all intercepts, vertices, and asymptotes on graphs.

• For word problems, define variables clearly and set up equations before solving.