Functions and their graphs

Graphing rational functions Video transcript Voiceover:

Functions and their graphs

Algebraic functions are functions which can be expressed using arithmetic operations and whose values are either rational or a root of a rational number. Now, we will be dealing with transcendental functions. Transcendental functions return values which may not be expressible as rational numbers or roots of rational numbers.

Algebraic equations can be solved most of the time by hand. Transcendental functions can often be solved by hand with a calculator necessary if you want a decimal approximation. However when transcendental and algebraic functions are mixed in an equation, graphical or numerical techniques are sometimes the only way to find the solution.

The simplest exponential function is: That is a pretty boring function, and it is certainly not one-to-one.

Recall that one-to-one functions had several properties that make them desirable. They have inverses that are also functions. They can be applied to both sides of an equation. Here are some properties of the exponential function when the base is greater than 1.

The graph is asymptotic to the x-axis as x approaches negative infinity The graph increases without bound as x approaches positive infinity The graph is continuous The graph is smooth What would the translation be if you replaced every x with -x? It would be a reflection about the y-axis.

We also know that when we raise a base to a negative power, the one result is that the reciprocal of the number is taken. Properties of exponential function and its graph when the base is between 0 and 1 are given. The graph is asymptotic to the x-axis as x approaches positive infinity The graph increases without bound as x approaches negative infinity The graph is continuous The graph is smooth Notice the only differences regard whether the function is increasing or decreasing, and the behavior at the left hand and right hand ends.

Translations of Exponential Graphs You can apply what you know about translations from section 1. It will not change whether the graph goes without bound or is asymptotic although it may change where it is asymptotic to the left or right.

By knowing the features of the basic graphs, you can apply those translations to easily sketch the new function. You should now add the exponential graph from the front cover of the text to the list of those you know. The limit notation shown is from calculus.

The limit notation is a way of asking what happens to the expression as x approaches the value shown. The limit is the dividing line between calculus and algebra. Calculus is algebra with the concept of limit.

People always have this dread of calculus that I can't understand. The calculus itself is easy. The reason people don't do well in calculus is not because of the calculus, but because of they are poor at algebra. The value for e is approximately 2. Here is a slightly more accurate, but no more useful, approximation.

On the TI-8x calculators, it is on the left side as a [2nd] [Ln]. The exponential function with base e is sometimes abbreviated as exp.Common Functions Reference. Here are some of the most commonly-used functions, and their graphs. Graphs of Basic Functions There are six basic functions that we are going to explore in this section.

We will graph the function and state the domain and range of each function. The six basic functions are the following: 1. () 2. () 3. The graph of the function is the graph of reflected in the x-axis.

Vertical stretching and shrinking: If is a real number, the graph of is the graph of stretched vertically by for or shrunk vertically by for. Graphing Rational Functions Date_____ Period____ Identify the points of discontinuity, holes, vertical asymptotes, x-intercepts, and horizontal asymptote of each.

The function concept is one of the most important ideas in study of mathematics beyond the elementary level requires a firm understanding of a basic list of elementary functions, their properties, and their graphs.

Characteristics of Functions and Their Graphs. A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time.

Functions and their graphs

In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions.

Functions and Graphs