Graph each equation. y=-(1)/(x)( Let x=-2,-1,-(1)/(2),-(1)/(3), (1)/(3), (1)/(2), 1, and 2 .)

Name: Problem 54, Chapter 1: Algebra and Trigonometry
Uploaded: 2020-06-14T22:55:56-08:00
Duration: 4 min 38 s
Channel: Mary Brese
Description: Graph each equation. y=-(1)/(x)( Let x=-2,-1,-(1)/(2),-(1)/(3), (1)/(3), (1)/(2), 1, and 2 .)

Graph each equation. y=-(1)/(x)( Let ...

Key Concepts

Symmetry

The reciprocal function is an odd function, meaning it exhibits rotational symmetry about the origin. This property implies that f(-x) = -f(x) for all x in the domain, which is useful for predicting the behavior of the graph on both sides of the origin.

Domain and Range

The domain of the reciprocal function f(x) = 1/x is all real numbers except x = 0, as division by zero is undefined. Similarly, the range is all real numbers except y = 0. Understanding these restrictions helps in accurately sketching the graph and analyzing the function.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For the reciprocal function, there is a vertical asymptote at x = 0 since the function is undefined there, and a horizontal asymptote at y = 0, which the graph approaches as x becomes very large in magnitude.

Reciprocal Function

The reciprocal function is defined by f(x) = 1/x, meaning that for any non-zero x, the output is the multiplicative inverse of x. It exemplifies inverse variation, where the product of x and y is always 1, and it is a fundamental example of a rational function.

Graphing Functions

Graphing functions involves plotting points that satisfy the function's equation and understanding the overall shape of the curve. When graphing f(x) = 1/x, it is important to plot points in both the positive and negative regions to accurately capture its behavior and identify key features of the graph.

Transcript

00:01 I'm going to graph the equation y equals negative 1 over x by completing this table of values and then plotting the points.

00:10 The first thing i want to say is x cannot equal 0 because we cannot divide by 0.

00:16 We cannot use 0 in the denominator of a fraction.

00:19 So beginning with our input x equals negative 2, if we have negative 1 divided by negative 2, that simplifies to 1 half.

00:33 Next, we have negative 1, so that's negative 1 divided by negative 1 equals positive 1.

00:44 When we divide by a fraction, we have to multiply by the reciprocal.

00:50 So negative 1 divided by negative 1 half is actually negative 1 times negative 2, which is positive 2.

01:02 Negative 1 divided by negative 1 3 3.

01:06 Is equal to negative 1 times negative 3 is positive 3.

01:18 1 divided by 1 3rd is equal to 1 times 3.

01:24 Oh, i'm sorry, that's negative 1.

01:25 Divided by 1 3 3rd is negative 1 times 3 is negative 1.

01:37 1 divided by 1 half is the same as negative 1 times 2, which is negative 2.

01:46 Negative 1 divided by 1 is negative 1 and 1 divided by 2 negative 1 divided by 2 is negative 1 half and now we want to graph these points starting with negative 2 1 1 half so negative 2 is 2 to the left 1 half is just half of the unit up negative 1 1 1 so that's 1 unit to the left 1 unit up negative 1 1, 1, 1 ,5, positive 2, that's half a unit to the left, up 2.

02:29 Negative 1 3rd is 1 3rd to the left, up 3...

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