Quadratic Functions

Definition and Standard Form

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

• Standard Form: , where are real numbers and .

• The square function is the simplest quadratic function.

Examples:

Vertex Form and Transformations

Quadratic functions can also be written in vertex form:

• Vertex Form:

• The vertex is at .

• Transformations:

  • Horizontal shift by units

  • Vertical shift by units

  • : vertical stretch

  • : vertical compression

  • : opens upward (minimum)

  • : opens downward (maximum)

Example: has vertex and opens downward (maximum).

Key Features of Parabolas

• Vertex: The highest or lowest point, .

• Axis of Symmetry:

• x-intercepts: Solve

• y-intercept:

• Domain: (all real numbers)

• Range: if (minimum), if (maximum)

• Intervals of Increase/Decrease: Increasing to the right of the vertex, decreasing to the left (for ; reversed for )

Graphing Quadratic Functions

1. Identify the vertex

2. Find the axis of symmetry

3. Find x-intercepts by solving

4. Find the y-intercept by computing

5. Plot points and connect with a smooth curve

Example:

• Vertex:

• Axis of symmetry:

• x-intercepts: Solve

• y-intercept:

Converting Standard Form to Vertex Form (Completing the Square)

1. Factor from the first two terms:

2. Add and subtract inside the parentheses

3. Rewrite as a perfect square trinomial and simplify

4. Resulting form:

Example:

• Rewrite:

• Vertex form:

Understanding Polynomial Functions

Definition and Standard Form

• A polynomial function is a function of the form , where is a non-negative integer and all exponents are whole numbers.

• Standard form: Terms are ordered by descending powers of .

• Degree: The highest exponent .

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

• Domain: (all real numbers)

Example: (degree 4, leading coefficient 6)

Identifying Polynomial Functions

• Polynomials have only non-negative integer exponents.

• No negative or fractional exponents.

Example:

• is a polynomial (degree 2, leading coefficient 3)

• is a polynomial (degree 1, leading coefficient 1)

• is a polynomial (degree 3, leading coefficient 4)

• is not a polynomial (negative exponent)

Graphs of Polynomial Functions

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

• The degree and leading coefficient determine the end behavior.

End Behavior

• Describes what happens to as or .

• Determined by the leading term .

Zeros and Multiplicity

• Zeros (roots): Values of where (x-intercepts).

• Found by factoring and setting each factor to zero.

• Multiplicity: The number of times a zero occurs.

• If multiplicity is even, the graph touches the x-axis and turns around.

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

Example:

• , multiplicity 1 (crosses)

• , multiplicity 2 (touches)

• , multiplicity 3 (crosses)

Turning Points

• Points where the graph changes direction (local maxima or minima).

• The maximum number of turning points is , where is the degree.

Example: has at most 3 turning points.

Graphing Polynomial Functions

1. Determine end behavior from the leading term.

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

3. Find the y-intercept ().

4. Identify turning points (up to ).

5. Break the graph into intervals between zeros and asymptotes, plot points in each interval.

6. Connect points 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 .

• The domain is all real numbers except where .

• To write in lowest terms, factor numerator and denominator and cancel common factors (after finding the domain).

Example:

• Domain:

• Lowest terms: for

Asymptotes

• Asymptotes are lines that the graph approaches but does not cross (infinite behavior).

• Types: Vertical, Horizontal, and Removable (holes).

Vertical Asymptotes

• Found by setting the denominator (in lowest terms) equal to zero and solving for .

• Example: has a vertical asymptote at (after cancelling common factors).

Removable Discontinuities (Holes)

• Occur when a factor cancels from both numerator and denominator.

• Set the cancelled factor equal to zero to find the -value of the hole.

• Represented as open circles on the graph.

Example: has a hole at .

Horizontal Asymptotes

• Determined by comparing the degrees of numerator and denominator:

  • If degree numerator < degree denominator:

  • If degrees are equal:

  • If degree numerator > degree denominator: No horizontal asymptote (may have a slant asymptote)

Example: has as a horizontal asymptote.

Graphing Rational Functions

1. Factor numerator and denominator; find the domain by setting denominator .

2. Find holes by setting common factors to zero and cancelling.

3. Find x-intercepts by setting numerator to zero.

4. Find y-intercept by computing .

5. Find vertical asymptotes by setting denominator (in lowest terms) to zero.

6. Find horizontal/slant asymptotes by comparing degrees.

7. Break the graph into intervals between vertical asymptotes and zeros; plot a point in each interval.

8. Connect points, drawing the graph approaching asymptotes.

Transformations of Rational Functions

• Basic rational functions: and

• General transformation:

• Shifts:

  • Horizontal shift by units

  • Vertical shift by units

  • Reflection if

  • Stretch/compression by

Example: is shifted right 3 units and up 1 unit.

Summary Table: Asymptotes of Rational Functions

Key Formulas and Concepts

• Quadratic Formula:

• Vertex of : ,

• End Behavior: Determined by leading term

• Multiplicity: Even = touch, Odd = cross x-axis

• Horizontal Asymptote:

  • If ,

  • If ,

Practice and Application

• Identify the degree and leading coefficient of given polynomials.

• Find zeros and their multiplicities; state whether the graph crosses or touches the x-axis.

• Determine end behavior and maximum number of turning points.

• For rational functions, find domain, asymptotes, holes, and intercepts.

• Graph polynomial and rational functions using the steps outlined above.

Additional info: Some tables and graphs referenced in the original material were not fully visible; the above notes reconstruct the main concepts and procedures for Precalculus students based on standard academic content.