Graphing Overview

Introduction

Every family of functions has unique characteristics in terms of their symbolic, tabular, contextual, and graphical representations. After completing Algebra I and Algebra II, there should be a firm understanding of functions and function notation.

This unit will review families of functions learned in Algebra I and introduce new families of functions which will be presented in greater detail later in this course.

Linear Functions

Guided Practice

Absolute Value Functions

Absolute value is defined as a number’s distance from zero. Since distance is always positive or zero, absolute value functions will always produce positive or zero outputs.

Absolute value functions have equations in the form: . The graph has reflectional symmetry across a vertical line through the vertex. Absolute value functions can be piece-wise defined as two linear functions with different domains.

See the table and graph of a basic absolute value function below.

Here are some situations that can be modeled by absolute value functions:

• Distances in real life (time vs. distance).
• Exchanging currency (old currency vs. new).

Power Functions

A power function is a function where the variable has a power greater than one. Graphs of power functions will vary depending upon the value of the exponent.

Power functions have equations in the form: . There are patterns in the shapes of the graphs of power functions. Power functions take two forms: and . All power functions with even powers will graph similar to and power functions with odd powers will graph similar to .

See the table and graph of two basic power functions below.

	f(x) = x2 g(x) = x3

Square Root Functions

A square root function is a function that contains a variable under the square root symbol . Square and square root are inverse operations, so there will be similarities in their graphs.

Square root functions have equations in the form: or in exponential notation as . Graphs of square root functions increase quickly at first then flatten to a very slow rate of change.

See the table and graph of a basic square root function below.

f(x) = ?x

Here are some situations that can be modeled by square root functions:

• Power factor and electrical load distribution (power vs. load).
• Collision detection for polygon shaped objects (direction vs. size).
• Fly ball behind the plate (distance vs. time).

Video Lesson

Guided Practice

Exponential Functions

An exponential function is a function where a variable is found in the exponent. Exponential functions model many real life situations and are prevalent in news data.

Exponential functions are in the form: , where k is a constant. Graphs of exponential functions can grow or decay at startling rates. They turn quickly from a gradual increase to a steep vertical rise.

See the table and graph of a basic exponential function below.

Rational Functions

Prior to Algebra II, rational functions have not been presented. An overview is given here. This content will be fully developed in a later unit. Rational functions are the ratio of polynomials, in other words one polynomial divided by another.

Rational functions are in the form: . The graphs of rational functions will vary depending on the complexity of the polynomials in the numerator and denominator.

See the table and graphs of two of the most basic rational functions below.

Logarithmic Functions

Prior to Algebra II, logarithmic functions have not been presented. An overview is presented here. This content will be fully developed in a later unit. When given an exponential function, such as, it can be rewritten in logarithmic form as .

Logarithmic functions are in the form: ; . The graph of a logarithmic function will turn from a steep increase to an extremely slow gradual rise.

See the table and graph of a basic logarithmic function below.

• Richter scale for earthquakes (magnitude vs. energy).
• Measurement of sound (decibels vs. intensity).
• Cram studying (amount forgotten vs. time).

Interactive Activity