Quadratic Functions

Definition and Properties

A quadratic function is a polynomial of degree 2, typically written in standard form:

• Standard Form: , where and are real numbers.

• The square function is a basic quadratic function.

• Quadratic functions can also be written in vertex form for easier graphing:

• Vertex Form: , where is the vertex.

Properties of a Parabola:

• The graph of a quadratic function is a parabola.

• If , the parabola opens upward (vertex is a minimum).

• If , the parabola opens downward (vertex is a maximum).

• The axis of symmetry is the vertical line .

• Domain: All real numbers, .

• Range: If minimum at , ; if maximum at $k$, .

Graphing Quadratic Functions

To graph a quadratic function, follow these steps:

1. Identify the vertex: in vertex form, or use , in standard form.

2. Find the axis of symmetry: .

3. Find x-intercepts: Solve .

4. Find y-intercept: Compute .

5. Plot points and connect with a smooth curve.

Example:

• Vertex: [MAX]

• Axis of Symmetry:

• Domain:

• Range:

Transformations of Quadratic Functions

Quadratic functions can be transformed by shifting, stretching, or reflecting:

• Horizontal shift: shifts right by units.

• Vertical shift: shifts up by units.

• Vertical stretch/compression: stretches, compresses.

• Reflection: reflects over the x-axis.

Completing the Square

To convert standard form to vertex form:

1. Factor from the first two terms:

2. Add and subtract inside the parentheses.

3. Rewrite as

Example:

• Complete the square:

Understanding Polynomial Functions

Definition and Structure

A polynomial function is an expression of the form:

• Where is a non-negative integer, and all exponents are whole numbers.

• Degree: Highest exponent .

• Leading Coefficient: .

• Domain: Always .

Graphs of Polynomial Functions

• Graphs are continuous and smooth (no corners or breaks).

End Behavior

The end behavior of a polynomial function depends on its degree and leading coefficient:

• If degree is even and , both ends rise ( as ).

• If degree is even and , both ends fall ( as ).

• If degree is odd and , left end falls, right end rises.

• If degree is odd and , left end rises, right end falls.

Finding Zeros and Multiplicity

Zeros (roots) are values where . The multiplicity of a zero is the number of times a factor occurs.

• If multiplicity is even, the graph touches the x-axis at that zero.

• If multiplicity is odd, the graph crosses the x-axis at that zero.

Example:

• , multiplicity 1 (crosses)

• , multiplicity 2 (touches)

• , multiplicity 3 (crosses)

Turning Points

A turning point is where the graph changes direction. The maximum number of turning points is , where is the degree.

• Each turning point is a local maximum or minimum.

Graphing Polynomial Functions

To graph a polynomial function:

1. Determine end behavior from degree and leading coefficient.

2. Find x-intercepts (zeros) and their multiplicities.

3. Find y-intercept ().

4. Identify turning points.

5. Break the graph into intervals between known points and plot additional points.

6. Connect with a smooth, continuous curve.

Introduction to Rational Functions

Definition and Domain

A rational function is a function of the form , where and are polynomials and .

• Domain: All real numbers except where .

• To write in lowest terms, factor numerator and denominator and cancel common factors.

Example:

• Domain:

Asymptotes

Rational functions may have vertical, horizontal, or slant asymptotes.

• Vertical Asymptotes: Set denominator and solve for (after reducing to lowest terms).

• Horizontal Asymptotes: Determined by degrees of numerator and denominator:

Removable Discontinuities (Holes)

A hole occurs where a common factor cancels in numerator and denominator. To find holes:

1. Factor numerator and denominator.

2. Set common factor = 0 and solve for .

3. Holes are represented as open circles on the graph.

Graphing Rational Functions

To graph a rational function:

1. Factor and find domain (set denominator = 0).

2. Find holes (set common factors = 0).

3. Find x-intercepts (set numerator = 0).

4. Find y-intercept ().

5. Find vertical asymptotes (set denominator = 0).

6. Find horizontal/slant asymptotes.

7. Break graph into intervals and plot points in each.

8. Connect and draw curves approaching asymptotes.

Transformations of Rational Functions

Rational functions can be graphed using transformations of basic functions such as or :

• Horizontal shift: shifts right by units.

• Vertical shift: shifts up by units.

• Reflection: reflects over the x-axis.

Example:

• Vertical asymptote at

• Horizontal asymptote at

• Domain:

• Range:

Summary Table: Asymptotes of Rational Functions

Key Formulas and Concepts

• Quadratic Standard Form:

• Quadratic Vertex Form:

• Axis of Symmetry: or

• Polynomial Standard Form:

• Rational Function:

• Vertical Asymptote:

• Horizontal Asymptote: if degree numerator < denominator; if degrees equal

• Maximum Turning Points: for degree polynomial

Additional info: Some tables and practice problems were referenced but not fully visible; key concepts and procedures have been expanded for completeness.