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

Name: Problem 53, Chapter 1: Algebra and Trigonometry
Uploaded: 2020-06-14T22:54:03-08:00
Duration: 4 min 40 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 .)

step 1 I need to graph the equation y equals 1 over x. That is also 1 divided by x depending on how you

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.

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.

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.

Transcript

00:02 I need to graph the equation y equals 1 over x.

00:07 That is also 1 divided by x depending on how you look at it.

00:13 We're using the given values for x.

00:16 So our first input value is negative 2.

00:18 So that's 1 divided by negative 2, which is negative 1 half.

00:24 So we've got the ordered pair negative 2, negative 1 half.

00:30 Next we're using negative 1, so that's 1 divided by negative 1, which is negative 1.

00:34 1.

00:38 When we use a fraction and we are dividing by a fraction, so that's 1 divided by negative 1 half, the rule of dividing by a fraction is to multiply by the reciprocal, so that's 1 times negative 2 is negative 2.

01:01 1 divided by negative 1 3 is the same as 1 times negative 3.

01:16 1 divided by 1 3rd is the same as 1 times 3.

01:20 Three, one divided by one half is the same as one times two.

01:38 One divided by one is one, and one divided by two is one half.

01:54 Now we need to very carefully graph these points.

01:59 Our first point is at negative two, negative one half.

02:03 So that's two to the left and one half of a unit down.

02:09 Negative one, negative one, is to the left one down one.

02:15 Negative 1 half, negative 2 is to the left 1 half down 2.

02:26 We don't have x equals 0 because we cannot put 0 in the denominator...