Function Review

Understanding Functions

A function is a rule that assigns to each input exactly one output. If we write y = f(x), then y is the value we obtain when we substitute x into the function.

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

• Range: The set of all possible output values (y-values) that the function can produce.

• Expanding/FOIL: The process of multiplying two binomials using the First, Outer, Inner, Last method.

Example: If f(x) = x^2 + 2x + 1, then the domain is all real numbers, and the range is all real numbers greater than or equal to 0.

Parabolas

Key Features of Parabolas

A parabola is the graph of a quadratic function of the form y = ax^2 + bx + c. Understanding its features is essential for graphing and analysis.

• x-intercepts: Points where the parabola crosses the x-axis. Found by solving ax^2 + bx + c = 0.

• y-intercept: The point where the parabola crosses the y-axis. Found by evaluating y = c when x = 0.

• Vertex: The maximum or minimum point of the parabola. The x-coordinate is given by .

• To find the y-coordinate of the vertex, substitute the x-value into the original equation.

• Equation from Graph: If you know the x-intercepts, y-intercept, and vertex, you can reconstruct the equation of the parabola.

• Domain: For all quadratic functions, the domain is all real numbers.

• Range: If a > 0, the range is where k is the y-coordinate of the vertex. If a < 0, the range is .

Example: For y = 2x^2 - 4x + 1:

• Vertex x-coordinate:

• Vertex y-coordinate:

• Vertex: (1, -1)

• Domain: All real numbers

• Range:

Rational Functions and Their Graphs

Graphing and Simplifying Rational Functions

A rational function is a function of the form , where P(x) and Q(x) are polynomials and Q(x) \neq 0.

• Graphing: To graph a rational function, identify intercepts, asymptotes, holes, and the general shape.

• Simplifying Rational Expressions: Factor numerator and denominator, then cancel common factors. Always check for restrictions on the domain before canceling.

• Domain: All real numbers except those that make the denominator zero.

• Asymptotes:

  • Vertical asymptotes: Occur at values of x where the denominator is zero and the factor does not cancel.

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

• Holes: Occur where a factor cancels from both numerator and denominator, indicating a removable discontinuity.

Example: For :

• Factor:

• Vertical asymptotes at and

• No holes since no common factors cancel

• Domain: All real numbers except

Example: For :

• Domain: All real numbers except and

• Vertical asymptotes at and

Polynomial Functions: Roots and Theorems

Roots, Zeros, and Imaginary Numbers

The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero. For real polynomials, if a i is a root, then -a i is also a root (complex conjugate pairs).

• Root/Zero: A value r such that f(r) = 0.

• Real roots: Correspond to x-intercepts on the graph.

• Imaginary roots: Occur in conjugate pairs for polynomials with real coefficients.

Key Theorems and Techniques

• Descartes' Rule of Signs: Predicts the possible number of positive and negative real roots of a polynomial by counting sign changes in the coefficients.

• Rational Root Theorem: Provides a list of possible rational roots of a polynomial equation as , where p divides the constant term and q divides the leading coefficient.

• Theorem on Bounds: Helps to determine upper and lower bounds for the real roots of a polynomial.

• Synthetic Division: A shortcut method for dividing a polynomial by a linear factor of the form x - c. Useful for testing possible roots and factoring polynomials.

Example: To use the Rational Root Theorem for , possible rational roots are .

Calculator and Test Policies

Test Preparation and Allowed Materials

• Graphing calculators are expected for use during tests.

• One note sheet (one side of a full-sized sheet) is allowed, containing only formulas and written concepts (no homework solutions).

• Write your name clearly on your note sheet and turn it in after the test.

• Tests are 50 minutes long; practice working at this pace.

• No headphones, smart watches, or other electronic devices are allowed during the test.

Study Strategies

• Attend class regularly and complete all assignments on time.

• Take notes on homework problems and topics that need review.

• Complete the review assignment and practice problems by hand.

Additional info: For more practice, refer to specific homework problems mentioned (e.g., HW 1 #9, #10; HW 2 #6, #10; HW 3 #1) for comprehensive coverage of these concepts.