Quadratic Functions

Understanding Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically written in the form . They model many real-world phenomena, such as the trajectory of a ball.

• Standard Form:

• Vertex Form: , where is the vertex of the parabola

• Factored Form: , where and are the roots

Key Properties

• Vertex: The highest or lowest point of the parabola, found at

• Axis of Symmetry: The vertical line

• Direction: If , the parabola opens upward; if , it opens downward

• Y-intercept: The point

• X-intercepts (Roots): Solutions to

Example: Ball Trajectory

The height (in meters) of a ball as a function of horizontal distance (in meters) is given by:

• Initial Height: meters

• Maximum Height: Occurs at meters

• Maximum Value: Substitute into to find the maximum height

• Domain: Values of for which (the ball is above the ground)

• Range:

Concavity

• If , the function is concave up (U-shaped)

• If , the function is concave down (n-shaped)

Graphing Quadratic Functions

• Plot the vertex, axis of symmetry, and intercepts

• Sketch the parabola based on the direction of opening

Polynomial Functions

General Properties

Polynomial functions are sums of terms of the form , where is a non-negative integer.

• Degree: The highest power of

• Leading Coefficient: The coefficient of the highest degree term

• End Behavior: Determined by the degree and leading coefficient

• Zeros: Values of where the function equals zero

Example: Cubic Polynomial

• Degree: 3 (cubic)

• Zeros:

• End Behavior: As , ; as ,

Graphing Polynomials

• Plot zeros and y-intercept

• Determine end behavior from degree and leading coefficient

• Sketch the curve, noting turning points (maximum of for degree )

Exponential Functions

Definition and Properties

An exponential function has the form , where and , .

• Growth: If , the function grows as increases

• Decay: If , the function decays as increases

• Y-intercept:

• Horizontal Asymptote:

Growth Factor and Rate

• Growth Factor: , where is the growth rate (as a decimal)

• Percent Rate of Change:

Example:

• For a 5% growth rate:

• For a 10% decay rate:

Matching Equations to Graphs

• Positive and : Increasing exponential

• Negative : Reflection over the x-axis

• : Decreasing exponential

Table Example

Additional info: This table shows values of an exponential function for various x-values.

Applications

• Compound interest:

• Continuous compounding:

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. It is written as , which means .

• Common Logarithm: Base 10,

• Natural Logarithm: Base ,

Properties of Logarithms

Solving Logarithmic and Exponential Equations

• Rewrite exponentials as logarithms and vice versa

• Example:

• Example:

Applications

• Half-life:

• Solving for time in exponential growth/decay

Order of Magnitude and the Number

Understanding

The number is the base of the natural logarithm and arises in continuous growth and decay processes.

• Order numbers: from smallest to largest

• Continuous growth formula:

Compound Interest and Effective Annual Rate

Formulas

• Compound Interest:

• Continuous Compounding:

• Effective Annual Rate (EAR):

Example:

• Find the value of a A = 70e^{0.042 \times 4}$

Summary Table: Exponential and Logarithmic Equations

Additional info: Table summarizes key logarithmic properties and conversions.