Sketch the graph of the function by first making a table of values. g(x)=(2)/(x^2)

Name: Problem 21, Chapter 3: Algebra and Trigonometry
Uploaded: 2020-09-08T03:55:44-08:00
Duration: 5 min 3 s
Channel: Nick Johnson
Description: Sketch the graph of the function by first making a table of values. g(x)=(2)/(x^2)

Sketch the graph of the function by first making a table of...

Key Concepts

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually reaches. In the context of rational functions, vertical asymptotes typically arise where the denominator is zero, indicating points of discontinuity, while horizontal or slant asymptotes describe the long-term behavior of the function as the input values grow large in magnitude.

Domain

The domain of a function consists of all input values for which the function is defined. For rational functions, the domain is particularly important since values that make the denominator zero must be excluded to prevent undefined expressions.

Rational Functions

Rational functions are expressed as the ratio of two polynomials. They often exhibit distinctive characteristics such as asymptotic behavior and discontinuities. Understanding these traits is essential for accurately sketching their graphs.

Table of Values

A table of values is a systematic way to organize selected inputs (x-values) and their corresponding outputs (f(x)-values). This helps in constructing the graph by providing concrete points through which the function passes, making it easier to capture the function’s behavior especially around areas of interest.

Graphing Functions

Graphing functions involves plotting a set of points based on input-output pairs and then drawing the curve that connects these points. This process helps visualize the behavior of the function, identify key features like intercepts and turning points, and understand overall trends.

Transcript

00:01 Okay, so here we have the function, g of x is equal to 2 over x squared.

00:05 So if we go ahead and make a table, so we have x and we have g of x.

00:11 You put in some negative values.

00:13 I don't know how we start it, maybe, i don't know, negative 3.

00:15 So when x is negative 3, what is g of x? well, negative 3, you always put in x in processes, right? because negative 3 squared, right, this is actually negative 9, because you have a negative 1 in front of, this is the opposite of 3 squared, would be negative 9.

00:34 But you plug in negative 3 to g of x.

00:36 You're always plugging in the entire thing.

00:38 It always goes in parentheses.

00:40 So we have 2 over negative 3 squared.

00:45 So that be 2 over positive 9.

00:48 So when x is negative 3, g of x is 2 over 9 or 2 9ths.

00:55 When x is, let's say, negative 2, well, you would get 2 over negative 2 squared.

01:04 It's got to be 2 over 4, which would be 2 4s, which is 1⁄2.

01:10 So when x is negative 2, g of x is 1 half.

01:15 When x is, well, negative 1, right? negative 1 squared is 1.

01:22 So g of x would be 2 over 1, which would be positive 2, right? and we see that g of x is always here.

01:30 Negative number i put in, g of x is always going to be a positive, right? when g of x was one, g of x was two, right? if you put in, let's say, a number less than one, but bigger than zero, so a fraction, let's say maybe i pick, i don't know, one -fourth, let's say, what would g of x be? so then we would have two over, what did i pick? i picked one -fourth...

Please give Ace some feedback