Linear Functions

Definition and General Form

Linear functions are algebraic functions whose graphs are straight lines. They model relationships with a constant rate of change between two variables.

• General Form:

• Name: This is called the slope-intercept form.

• Graph: The graph is a straight line where m is the slope and b is the y-intercept.

• Slope Formula: , where .

• Rate of Change: in is the slope. It means the change in per unit change in .

• Constant Rate: Linear functions have a constant rate of change in and .

• Nonlinear Functions: Functions that do not have a constant rate of change are called nonlinear functions.

Behavior of Linear Functions

The behavior of a linear function depends on the value of the slope :

• If , then increases left-to-right over its domain.

• If , then decreases left-to-right over its domain.

• If , then is constant left-to-right over its domain.

Identifying Linear vs. Nonlinear Functions from Data

To determine if a function is linear or nonlinear from a table of values, check if the rate of change () is constant. If so, the function is linear; otherwise, it is nonlinear.

Example: For the second table, for each step, so the function is linear. The equation is .

Applications of Linear Functions

Linear functions are used to model real-world relationships with constant rates of change. Common applications include cost, revenue, and depreciation problems.

• Solving Equations: To find where two linear functions are equal, set and solve for .

• Inequalities: To solve , solve for and express the solution in interval notation.

Depreciation Example

A company buys a computer for V(x)$ can be written in slope-intercept form:

• When ,

• When ,

• Slope:

• Value Function:

Domain of a Linear Model

The implied domain of is , since the computer's age cannot be negative or exceed 3 years.

Evaluating and Solving Linear Models

• To find after years:

• To find when : year

Quadratic Functions

General Form and Properties

Quadratic functions are polynomial functions of degree 2. Their graphs are parabolas, which open upward or downward depending on the leading coefficient.

• General Form: ,

• Graph: The graph is a parabola.

• If , the parabola opens upward.

• If , the parabola opens downward.

• Vertex Formula: The vertex is given by ,

• Axis of Symmetry:

• y-intercept:

• x-intercepts: Solve

Domain and Range

• Domain: All real numbers,

• Range: If , ; if ,

To find the range from a graph, read from the lowest to the highest point (bottom to top).

Increasing and Decreasing Intervals

For :

• Decreasing on

• Increasing on

Vertex Form of a Quadratic Function

The vertex form is , where is the vertex.

Applications of Quadratic Functions

Quadratic functions are used to model situations involving maximum or minimum values, such as projectile motion, area optimization, and revenue problems.

• Example: Find a quadratic function with vertex and y-intercept .

• Use vertex form:

• Plug in , to solve for .

Graphing Quadratic Functions

To graph :

• Find the vertex using

• Find another point by substituting a value for

• Determine domain, range, and intervals of increase/decrease

Building Quadratic Models

Translating Real-World Problems

Quadratic models are constructed by translating real-world scenarios into quadratic equations, then solving for optimal values (maximum or minimum).

• Identify variables and relationships

• Write the quadratic equation representing the situation

• Find the vertex for optimal values

Applications of Quadratic Models

• Fencing Problem: 2000 feet of fencing is used to enclose three sides of a rectangular field. Find the maximum area.

• Revenue Problem: Given a demand equation , , express revenue as a function of and find the maximum revenue.

• Suspension Bridge Problem: Find the height of cables at a certain distance from the center, given the parabolic shape of the cables.

Example (Revenue): If , the maximum revenue occurs at the vertex .

Solving Quadratic Models

• Find the vertex to determine maximum or minimum values

• Solve for or as required by the context

• Express answers in appropriate units and context

Additional info: These notes cover the core concepts and applications of linear and quadratic functions, including modeling, graphing, and solving real-world problems, as outlined in a typical college algebra curriculum.