Equations and Inequalities

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically written in the form . The solutions (roots) can be found using factoring, completing the square, or the quadratic formula.

• General Form:

• Given Solutions: If the solutions are and , the equation can be written as .

• Example: For solutions (2, -1), the equation is or .

Linear Inequalities

Linear inequalities involve expressions with a variable to the first power. The solution is often represented in interval notation.

• Solving: Isolate the variable using algebraic operations, remembering to reverse the inequality sign when multiplying or dividing by a negative number.

• Interval Notation: Expresses the set of solutions, e.g., .

• Example: Solve .

Functions and Their Graphs

Definition of a Function

A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: The set is not a function because the input 1 maps to two different outputs (11 and 22).

Tables and Functions

Tables can be used to determine if a relation is a function by checking if each input value corresponds to only one output value.

• Domain:

• Range:

• Function? Yes, if each x-value is unique.

Linear and Quadratic Functions

Equations of Lines

The equation of a line can be written in slope-intercept form () or point-slope form ().

• Slope (m): The rate of change of the line.

• Example: For slope and passing through , use point-slope form first, then solve for .

Equations of Circles

The standard form for the equation of a circle with center and radius is:

• Example: Center , :

Transformations of Functions

Square Root and Absolute Value Functions

Transformations include translations, reflections, stretches, and compressions.

• Vertical Stretch: is a vertical stretch by 3 and a horizontal shift left by 4 units of .

• Absolute Value: is a transformation of .

Function Operations and Composition

Function Composition

Given two functions and , the composition means to substitute into .

• Example: If , , then .

Quadratic Functions

Vertex Form

The vertex form of a quadratic function is , where is the vertex.

• Example: For vertex and same shape as , .

Polynomial and Rational Functions

Graphing Polynomial Functions

Polynomial functions can be graphed by finding their zeros, end behavior, and plotting key points.

• Example: is a quadratic with zeros at and .

Dividing Polynomials

Polynomial long division is used to divide one polynomial by another, resulting in a quotient and a remainder.

• Example: Divide by .

Zeros of Polynomial Functions

To find all zeros of a polynomial, use factoring, the Rational Root Theorem, or the quadratic formula.

• Example: If is a zero of , factor to find other zeros.

End Behavior of Polynomials

The end behavior of a polynomial function is determined by the degree and leading coefficient.

• Leading Coefficient Test: For , as , ; as , (since degree is even and leading coefficient is positive).

Rational Functions

Graphing Rational Functions

Rational functions are of the form . Key features include vertical and horizontal asymptotes, holes, and intercepts.

• Vertical Asymptotes: Values of where and .

• Horizontal Asymptotes: Determined by the degrees of and .

• Example:

Asymptotes and Intercepts

• Vertical Asymptote: Set denominator equal to zero and solve for .

• Horizontal Asymptote: Compare degrees of numerator and denominator.

• x-intercept: Set numerator equal to zero and solve for .

Summary Table: Key Features of Functions

Additional info: This guide covers core College Algebra topics including equations, inequalities, functions, graphing, transformations, and polynomial/rational function analysis, as reflected in the midterm questions.