Polynomial Functions

Quadratic Functions

Quadratic functions are a special class of polynomial functions of degree 2. Their graphs are parabolas, and they can be written in several forms, including standard and vertex form. Understanding their properties is essential for graphing and analyzing their behavior.

• Definition: A quadratic function is a polynomial of degree 2, typically written as , where .

• Standard Form:

• Vertex Form: , where is the vertex.

• Square Function: is the basic quadratic function.

• Parabola Properties:

  • Vertex: The point ; can be a minimum (if ) or maximum (if ).

  • Axis of Symmetry:

  • x-intercepts: Solutions to

  • y-intercept:

  • Domain: All real numbers,

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

  • Increasing/Decreasing: Decreasing to the left of the vertex, increasing to the right (for )

• Example: has vertex (maximum), axis of symmetry , domain , range .

Vertex Form & Transformations

The vertex form of a quadratic function allows for easy identification of the vertex and understanding of transformations.

• Vertex Form:

• Transformations:

  • Horizontal Shift: units right if , left if

  • Vertical Shift: units up if , down if

  • Vertical Stretch: stretches the graph

  • Vertical Compression: compresses the graph

  • Direction: opens upward, opens downward

• Example: has vertex , opens upward.

Graphing Quadratic Functions

To graph a quadratic function, follow these steps:

1. Identify the vertex

2. Find the axis of symmetry

3. Find x-intercepts by solving

4. Find y-intercept by computing

5. Plot points and connect with a smooth curve

Completing the Square

To convert a quadratic function from standard form to vertex form, use the method of completing the square.

1. Factor from the first two terms:

2. Add and subtract inside the parentheses

3. Move the subtracted term outside, multiplied by

4. Rewrite as

• Example: Complete the square: Vertex:

Understanding Polynomial Functions

Definition and Properties

A polynomial function is an expression involving only non-negative integer exponents of the variable, with real coefficients.

• General Form:

• Degree: The highest exponent

• Leading Coefficient:

• Domain: All real numbers,

• Standard Form: Terms in descending order of degree

• Graphs: Continuous and smooth (no corners or breaks)

• Example: is a degree 4 polynomial with leading coefficient 6.

Identifying Polynomial Functions

To determine if a function is a polynomial:

• Check for only non-negative integer exponents

• No variables in denominators or under radicals

• Combine like terms and write in standard form

• Example: is a polynomial of degree 2, leading coefficient 3.

End Behavior of Polynomial Functions

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

• Even Degree: Both ends go up if , both go down if

• Odd Degree: Left and right ends go in opposite directions; right rises if , falls if

Finding Zeros and Multiplicity

The zeros (roots) of a polynomial are the values of where . The multiplicity of a zero is the number of times a factor appears.

• Finding Zeros: Factor the polynomial and set each factor equal to zero.

• Multiplicity: If a zero has even multiplicity, the graph touches (bounces off) the x-axis; if odd, it crosses.

• Example: Zeros: (multiplicity 1, crosses), (multiplicity 2, touches), (multiplicity 3, crosses)

Turning Points

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

• Local Maximum/Minimum: Each turning point is either a local maximum or minimum.

• Example: has degree 4, so at most 3 turning points.

Graphing Polynomial Functions

To graph a polynomial function, follow these steps:

1. Determine end behavior from degree and leading coefficient

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

3. Find y-intercept by computing

4. Identify turning points (maximum )

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

6. Connect with a smooth, continuous curve

• Example: Degree: 3 (odd), leading coefficient: 2 (positive) End behavior: Left falls, right rises Zeros: Solve y-intercept: Maximum turning points: 2

Summary Table: Key Properties of Polynomial Functions

Additional info:

• Some tables and practice problems in the original notes were incomplete; academic context and examples were added for clarity.

• All formulas are provided in LaTeX format for mathematical clarity.