4. Polynomial Functions

Quadratic Functions

Quadratic functions are a fundamental class of polynomial functions, characterized by their degree of 2. Their graphs are parabolas, and they are commonly written in standard or vertex form.

• Standard Form:

• Vertex Form:

Properties of a Parabola

• Vertex: The highest or lowest point of the parabola, given by in vertex form or in standard form.

• Axis of Symmetry: The vertical line passing through the vertex, or .

• Direction: Opens upward if , downward if .

• Domain: All real numbers, .

• Range: if ; if .

• Increasing/Decreasing: The function decreases to the vertex and increases after (if ), or vice versa (if ).

Example

• For :

  • Vertex: ,

  • Axis of Symmetry:

  • Opens upward ()

Vertex Form & Transformations

The vertex form allows for easy identification of transformations:

• Horizontal Shift: units right if , left if

• Vertical Shift: units up if , down if

• Vertical Stretch/Compression: stretches, compresses

• Reflection: If , reflects over the x-axis

Converting Standard Form to Vertex Form (Completing the Square)

• Given , complete the square to rewrite in vertex form.

• Example:

Graphing Quadratic Functions

1. Find the vertex

2. Find the axis of symmetry

3. Find y-intercept

4. Find x-intercepts by solving

5. Plot points and sketch the parabola

Understanding Polynomial Functions

Polynomial functions are expressions of the form , where is a non-negative integer and coefficients are real numbers.

• Degree: The highest power of with a nonzero coefficient.

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

• Standard Form: Terms are written in descending order of degree.

Identifying Polynomial Functions

• All exponents must be non-negative integers.

• Coefficients must be real numbers.

• Examples:

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

  • is not a polynomial (negative exponent)

Graphs of Polynomial Functions

• Polynomial functions are continuous and smooth (no breaks, holes, or sharp corners).

• The domain is always .

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as or . It is determined by the degree and leading coefficient.

Example

• (degree 3, leading coefficient 2): Left end falls, right end rises.

Zeros and Multiplicity

Zeros (roots) of a polynomial are the values of for which . The multiplicity of a zero is the number of times its corresponding factor appears in the factorization.

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

• If a zero has even multiplicity, the graph touches but does not cross the x-axis at that zero.

Example

• Zero at , multiplicity 2 (touches x-axis) Zero at , multiplicity 3 (crosses x-axis)

Turning Points

A turning point is where the graph changes direction from increasing to decreasing or vice versa. The maximum number of turning points of a polynomial of degree is .

• For (degree 4): Maximum 3 turning points.

Graphing Polynomial Functions

To graph a polynomial function, follow these steps:

1. Determine end behavior using degree and leading coefficient.

2. Find and plot y-intercept ().

3. Find and plot x-intercepts (solve ).

4. Determine multiplicity of each zero (decide if the graph crosses or touches at each zero).

5. Find turning points (maximum for degree ).

6. Plot additional points as needed for accuracy.

7. Connect points with a smooth, continuous curve.

Example

• For :

  • Degree: 3 (odd), leading coefficient: 1 (positive)

  • End behavior: Left falls, right rises

  • Y-intercept:

  • X-intercepts:

  • Turning points: Maximum 2

Summary Table: Key Properties of Polynomial Functions

Additional info: Practice problems and step-by-step graphing guides are included throughout the notes to reinforce understanding and application of these concepts.