Comprehensive Study Guide: College Algebra Core Concepts and Applications

Polynomial Functions

Identifying Degree and Leading Coefficient

Polynomial functions are classified by their degree (the highest power of x) and the sign of their leading coefficient (the coefficient of the highest degree term). These characteristics determine the end behavior of the graph.

• Degree: The highest exponent of x in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• End Behavior: For even degrees, both ends of the graph go in the same direction; for odd degrees, they go in opposite directions. The sign of the leading coefficient determines whether the graph rises or falls as x approaches infinity.

• Example: For $f(x) = 2x^3 - 5x^2 + x - 7$, the degree is 3 (odd), and the leading coefficient is 2 (positive).

Finding Intercepts and Multiplicity

Intercepts and multiplicities provide information about where the graph crosses or touches the axes.

• x-intercepts: Set $f(x) = 0$ and solve for x.

• y-intercept: Set $x = 0$ and solve for $f(0)$.

• Multiplicity: The number of times a particular root occurs. If the multiplicity is even, the graph touches the x-axis and turns around; if odd, it crosses the axis.

• Example: For $f(x) = (x-2)(x+1)^3$, x-intercepts are at $x=2$ (multiplicity 1) and $x=-1$ (multiplicity 3).

Factoring Polynomials

Factoring expresses a polynomial as a product of its linear or irreducible quadratic factors.

• Lowest Degree: The factored form should use the smallest possible degree for each factor.

• Example: $f(x) = x^3 - 4x^2 + 4x$ factors as $x(x-2)^2$.

Solving Equations

Linear and Rational Equations

Solving equations involves isolating the variable using algebraic operations.

• Linear Equations: Equations of the form $ax + b = c$.

• Rational Equations: Equations involving fractions with variables in the denominator. Multiply both sides by the least common denominator (LCD) to clear fractions.

• Example: Solve $\frac{4}{x} - 2 = \frac{6}{x+2}$ by finding the LCD and solving for x.

Variation and Applications

Direct and Inverse Variation

Variation describes how one quantity changes in relation to another.

• Direct Variation: $y = kx$, where k is the constant of variation.

• Inverse Variation: $y = \frac{k}{x}$.

• Example (Ohm's Law): $I = \frac{k}{R}$, where I is current and R is resistance.

Word Problems and Applications

• Hooke's Law: $D = kW$, where D is distance stretched and W is weight.

• Maximum Weight Supported: $W = kH^2$, where W is weight and H is height.

• Population Models: $N(t) = \frac{40 + 20t}{1 + 0.03t}$ models population over time.

Rational Functions and Asymptotes

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches.

• 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.

• Example: For $f(x) = \frac{5x^2}{x^2-4}$, vertical asymptotes at $x=2$ and $x=-2$; horizontal asymptote at $y=5$.

Transformations of Functions

Base Functions and Transformations

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

• Base Functions: $y = \frac{1}{x}$ (reciprocal), $y = \frac{1}{x^2}$ (reciprocal squared).

• Transformations: Include translations (shifts), reflections, stretches, and compressions.

• Example: $f(x) = \frac{1}{x-5}$ is a horizontal shift of $y = \frac{1}{x}$ right by 5 units.

Function Operations and Composition

Function Composition

Composing functions involves applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• Domain: The set of all x-values for which the composition is defined.

• Example: If $f(x) = \frac{1}{x}$ and $g(x) = x+7$, then $(f \circ g)(x) = \frac{1}{x+7}$.

Inverse Functions

Finding and Interpreting Inverses

The inverse of a function "undoes" the action of the function. A function has an inverse if it is one-to-one (passes the horizontal line test).

• Finding the Inverse: Swap x and y in the equation and solve for y.

• Graphical Interpretation: The graph of the inverse is a reflection over the line $y = x$.

• Example: For $f(x) = \sqrt{x} + 1$, the inverse is $f^{-1}(x) = (x-1)^2$.

Tables and Numeric Representations

Numeric Representation of Functions and Inverses

Tables can be used to represent functions and their inverses numerically.

x	f(x)
-10	-6
-6	-2
-1	1
3	7
7	15

To find $f^{-1}(7)$, look for x such that $f(x) = 7$; here, $x = 3$.

Sample Table for Inverse

x	f(x)
2	-10
-1	-4
0	2
1	3
2	5

To give a numeric representation of $f^{-1}$, swap the x and f(x) columns.

Summary Table: Types of Variation

Type	Equation	Description
Direct Variation	$y = kx$	y increases as x increases
Inverse Variation	$y = \frac{k}{x}$	y decreases as x increases
Joint Variation	$y = kxz$	y varies directly with x and z

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Graph sketches and some specific values are not included due to the text-based format, but the principles for graphing and interpreting are described.