# 8-8 Function Junction - Concepts and Explanations

## Functions

Formal language and notation associated with the concept of function are introduced, with special attention given to step, piecewise, and polynomial functions.

### Example

A taxi cab owner charges his clients a fare based on the following rule: $5.00 for a distance of up to one mile and $2.00 for each additional mile or part of a mile. This rule is an example of a step function.

## Sequences

An arithmetic sequence is a sequence of numbers that has a constant rate of change. A geometric sequence is a sequence of numbers that has a constant factor. These sequences can be treated as linear and geometric functions, respectively, and students learn how to express them with rules in both recursive and closed forms.

### Example

An example of an arithmetic sequence would be:

s(n) = 1, 4, 7, 10, 13, ...

An example of a geometric sequence would be:

g(n) = 1, 3, 9, 27, 81, ...

## Translating and Stretching Functions

Students have previously worked with transformational geometry. In this Unit, students develop the connections between expressions for functions whose graphs are related by translation and one-dimensional stretching/shrinking. Students learn how to determine characteristics of graphs by looking at different forms of the equation.

### Example

Stretching the graph of *f(x)* = *x*^{2} toward the x-axis by a factor of 0.5, and then translating the graph down two units
and to the right three units results in the following graph, whose equation is *g(x)* = 0.5(*x* - 3)^{2} - 2.

## Completing the Square

Skill and understanding in the use of completing the square is developed to transform quadratic expressions to equivalent vertex forms.

### Example

To put the function *f(x)* = x^{2} + 4*x* - 4 into vertex form, a student could complete the square and obtain *f(x)* = (*x* + 2)^{2} - 8.

Often the vertex form gives us information about the graph of a function that is easier to see than when the function is written in standard form.

## Quadratic Formula

The Quadratic Formula is used for solving quadratic equations in the form *ax ^{2}* +

*bx*+

*c*= 0. The Quadratic Formula is also used to find complex numbers to provide solutions for cases where no real-number solutions exist.

### Example

Skill and understanding in the use of completing the square is developed to transform
quadratic expressions to equivalent vertex forms.

## Polynomial Functions

Analysis of polynomial functions and their graphs is extended to the study of the connection between expressions and graph properties, and then to develop operations with polynomial functions.

### Example

Some functions, called polynomial functions, have graphs that contain “hills” or “valleys”
that may not be maxima or minima. These hills and valleys are called *local maxima* and *local minima*, respectively. One example is the graph of *f(x)* = *x*^{3} + *x*^{2} - 6*x* + 2.