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Chapter 2: Problem 113

If the graph of \(y=m x+b\) passes through quadrants I, II, and IV, what do we know about the constants \(m\) and \(b\) ?

Short Answer

Expert verified
For the line to pass through quadrants I, II, and IV, the slope \(m\) must be negative and the y-intercept \(b\) must be positive.

Step by step solution

01

Identify Conditions of Line Passing Through Quadrants

For a line to pass through quadrants I, II, and IV, we must consider both the slope and y-intercept. Quadrant I requires positive y-values and positive x-values, Quadrant II requires positive y-values and negative x-values, and Quadrant IV requires negative y-values and positive x-values.
02

Determine Requirements for the Slope, \(m\)

For the line to pass through Quadrant II (where x is negative and y is positive) and also Quadrant IV (where x is positive and y is negative), the slope \(m\) must be negative. A negative slope will result in y increasing as x decreases and y decreasing as x increases.
03

Evaluate Requirements for the Y-intercept, \(b\)

The line must cross the y-axis where y is positive (to enter Quadrant I and II) and the x-axis where y is negative (to enter Quadrant IV). For this to occur, the y-intercept \(b\) must be positive. This aligns with the requirement for the graph to be in Quadrant I and II.

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness and is denoted by the letter \( m \) in the equation \( y = mx + b \).

It describes how much the \( y \)-value of a line changes for a unit change in the \( x \)-value. A positive slope means the line rises as you move from left to right on the graph. Conversely, a negative slope indicates the line will fall as you move right.

Remember:
  • A line that has no slope (i.e., \( m = 0 \)) is horizontal.
  • A vertical line has an undefined slope, since you can鈥檛 divide by zero (no change in x but a change in y).

Understanding slope is crucial for determining the direction in which your line moves across the coordinate plane.
Y-intercept
The y-intercept is the point where the line crosses the y-axis.

It is represented by \( b \) in the line equation \( y = mx + b \). This value is essential as it indicates the starting point of the line on the vertical axis, which is a key feature of the linear equation.

Keep in mind:
  • If \( b > 0 \), the line crosses the y-axis above the origin.
  • If \( b < 0 \), the line crosses below the origin.
  • If \( b = 0 \), the line passes through the origin.

The y-intercept can dramatically affect how a line fits a particular graph depending on its value.
Coordinate Planes
The coordinate plane is a two-dimensional surface defined by the x-axis and y-axis.

This is where you plot points and draw graphs.
The point where the x-axis and y-axis intersect is known as the origin, designated as \( (0, 0) \).

Key points about the coordinate planes:
  • The horizontal axis is the x-axis.
  • The vertical axis is the y-axis.

Understanding how to navigate and interpret the coordinate plane is a fundamental skill for graphing linear equations and identifying the position of lines and points.
Quadrants
The coordinate plane is divided into four sections known as quadrants. Each quadrant represents a different combination of positive and negative values for x and y coordinates.

Here's a simple breakdown of the quadrants:
  • Quadrant I: Both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: Both x and y are negative.
  • Quadrant IV: x is positive, y is negative.

This division helps specify the exact location of points and lines on a graph. Lines can traverse multiple quadrants, which is determined by their slope and y-intercept, as seen in the original exercise.

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Most popular questions from this chapter

Nurses. The demand for full-time registered nurses in the United States can be modeled by a linear function. In 2005 , approximately \(2,175,500\) nurses were needed. By the year \(2015,\) that number is expected to increase to about \(2,586,500\) (Source: National Center for Health Workforce Analysis) a. Let \(t\) be the number of years after 2000 and \(N\) be the number of full- time registered nurses needed in the U.S. Write a linear function \(N(t)\) to model the demand for nurses. b. Use your answer to part a to predict the number of full-time registered nurses that will be needed in \(2025,\) if the trend continues.

Fire Protection. City growth and the number of fires for a certain city are related by a linear equation. Records show that 113 fires occurred in a year when the local population was \(150,000\) and that the rate of increase in the number of fires was 1 for every \(1,000\) new residents. a. Using the variables \(p\) for population and \(F\) for fires, write an equation (in slope-intercept form) that the fire department can use to predict future fire statistics. b. How many fires can be expected when the population reaches \(200,000 ?\)

Write an equation for a linear function whose graph has the given characteristics. Passes through \((2,20),\) parallel to the graph of \(g(x)=8 x+1\)

Find \(h(5)\) and \(h(-2)\). \(h(x)=\frac{3}{x-4}\)

A student was asked to determine the slope of the graph of the line \(y=6 x-4 .\) If his answer is \(m=6 x,\) explain his error.
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