Quadratic Functions and X-Intercepts

Determining Values for No X-Intercepts

Quadratic functions are of the form . The x-intercepts are found by solving . A quadratic has no real x-intercepts if its discriminant is less than zero.

• Discriminant:

• No x-intercepts:

• Example: For , set and solve for .

Rational Functions and Domain

Identifying Graphs with Restricted Domains

A rational function is a ratio of polynomials, . The domain excludes values where . For example, if the domain is , then is excluded, indicating a vertical asymptote at $x = 5$.

• Vertical asymptote:

• Graph behavior: The function approaches infinity near .

• Example:

Polynomial Functions and Real Zeros

Determining Real Zeros

Real zeros of a polynomial are values of where . To check if a real zero satisfies a condition (e.g., no real zero greater than 3), analyze the roots using factoring or the Rational Root Theorem.

• Example:

• Condition: No real zero greater than 3

Graphing Rational Functions and Asymptotes

Finding Horizontal, Vertical, and Slant Asymptotes

Rational functions may have vertical, horizontal, or slant asymptotes. Vertical asymptotes occur where the denominator is zero. Horizontal asymptotes are determined by the degrees of numerator and denominator.

• Vertical asymptote: Set denominator to zero, e.g.,

• Horizontal asymptote: For , as ,

• Example:

Average Rate of Change

Calculating for a Given Interval

The average rate of change of from to is .

• Formula:

• Example: For , , :

Even and Odd Functions; Symmetry

Identifying Function Symmetry

A function is even if (symmetric about the y-axis), and odd if (symmetric about the origin).

• Example: is even because

Synthetic Division of Polynomials

Finding the Quotient

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form . The process involves using the coefficients and the value .

• Example: Divide by using synthetic division.

End Behavior of Polynomial Functions

Using the Leading Coefficient Test

The end behavior of a polynomial is determined by its leading term. For , the leading term is .

• Even degree, negative leading coefficient: Graph falls to the left and right.

• General rule: If degree is even and leading coefficient is negative, both ends fall.

Exponential and Logarithmic Functions

Graphing and Evaluating

Exponential functions have the form . Logarithmic functions are the inverse: .

• Example: is a decreasing exponential function.

• Logarithmic form:

• Example:

Properties of Logarithms

Combining Logarithms

Logarithm properties allow combining or separating logs:

• Product Rule:

• Example: If and , then (Additional info: If asked for in terms of and , context may require change of base formula.)

Solving Exponential and Logarithmic Equations

Finding Solutions

To solve equations involving logarithms or exponentials, isolate the variable and use properties of logarithms or exponentials.

• Example:

• Solution: or

• Therefore: or

Summary Table: Key Concepts

Additional info: Some problems involve graphical identification, which requires understanding of asymptotes, domain restrictions, and function transformations. All topics are foundational in College Algebra and align with standard curriculum chapters.