/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Sketch the graph of each functio... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 34

Sketch the graph of each function. \(x y+5=0\)

Short Answer

Expert verified
The graph of the function \(y = -x - 5\) is a straight line that crosses the y-axis at -5 and slopes downward from left to right.

Step by step solution

01

Rewrite the equation

The first task involves rewriting the equation in the form \(y = mx + b\) where \(m\) represents the slope and \(b\) is the y-intercept. From the given equation \(xy + 5 = 0\), re-arrange it to \(y = -x - 5\). So, the slope (\(m\)) is \( -1\) and the y-intercept (\(b\)) is \( -5\).
02

Identifying the points

We now have the slope and y-intercept. Let's draw the points on the graph. Starting from y-intercept which is at the coordinate (0, -5). Since the slope is -1, we move one step down for every step to the right from the y-intercept.
03

Sketch the graph

Draw a straight line through the plotted points. You'll notice that the line crosses the y-axis at -5 (the y-intercept) and that the line falls as it moves to the right, which reflects the negative slope of -1. So, this is the graph of the function \(y = -x - 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Linear Functions
Graphing a linear function means creating a visual representation of a linear equation on a coordinate plane. It involves plotting points that satisfy the given equation and then drawing a straight line through those points. A linear function appears as a straight line because it has a constant rate of change, known as the slope. By plotting the function using the slope and y-intercept, you can quickly visualize the graph.
  • To start graphing, rewrite the equation in slope-intercept form, if necessary.
  • Identify the y-intercept and plot it on the y-axis.
  • Use the slope to find another point, moving from the y-intercept.
  • Plot the second point and draw a line through the points.
A well-drawn graph can provide clear insights into the behavior of the function, such as its direction or points of intersection with the axes.
Equation of a Line
Understanding the equation of a line is crucial for graphing linear functions. The equation can take various forms, with the most common being the slope-intercept form, point-slope form, and standard form. Here, we'll focus on the slope-intercept form.
  • The general format is: \(y = mx + b\)
  • \(m\) represents the slope of the line, indicating its steepness and direction.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
In the given exercise, the equation \(xy + 5 = 0\) is transformed to \(y = -x - 5\), showing how the slope and y-intercept govern the line's position and orientation on the graph. Recognizing the equation of a line allows us to easily graph and analyze its properties.
Slope-Intercept Form
The slope-intercept form is a widely used method for expressing the equation of a line because it readily reveals key characteristics of the line. This form, \(y = mx + b\), directly provides the slope and y-intercept:
  • The slope \(m\) explains how the y-value changes for each unit increase in x. A positive slope means the line ascends while a negative slope indicates it descends.
  • The y-intercept \(b\) tells where the line will cross the y-axis. This is a starting point for plotting the line on a graph.
Using slope-intercept form makes graphing straightforward. In the exercise, converting the equation into this form helped us to identify that the slope is \(-1\) and the y-intercept is \(-5\). This information is crucial to sketching the line accurately on a coordinate plane.

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Most popular questions from this chapter

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