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Chapter 4: Problem 16

Graph a line with a negative slope and a positive \(y\) -intercept.

Short Answer

Expert verified
Choosing a negative slope of -2 and a positive y-intercept of 3, the equation of the line is \(y = -2x + 3\). To graph the line, find points on the line, such as (-2, 7), (0, 3), and (2, -1), then plot these points on the coordinate plane and connect them with a straight line.

Step by step solution

01

Choose a slope and y-intercept

Let's choose a negative slope of -2 and a positive y-intercept of 3. So, the equation of the line will be given by \(y = -2x + 3\).
02

Find other points on the line

To find other points on the line, we can choose some x-values and calculate the corresponding y-values using the equation of the line. Let's find the points for x = -2, 0, and 2. When x = -2: \(y = -2(-2) + 3 = 4 + 3 = 7\) So, the point (-2, 7) is on the line. When x = 0: \(y = -2(0) + 3 = 0 + 3 = 3\) So, the point (0, 3) is on the line (and also the y-intercept). When x = 2: \(y = -2(2) + 3 = -4 + 3 = -1\) So, the point (2, -1) is on the line.
03

Plot the points

On the coordinate plane, plot the points (-2, 7), (0, 3), and (2, -1).
04

Draw the line

Connect the points (-2, 7), (0, 3), and (2, -1) with a straight line to graph the complete line with a negative slope of -2 and a positive y-intercept of 3.

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Key Concepts

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

Negative Slope
A negative slope in a linear equation tells us that as we move from left to right on the graph, the line will descend. Essentially, the line goes downhill. This happens because, with a negative slope, the y-value decreases as the x-value increases. This is an important concept in graphing linear equations because it helps us predict the direction of the line.

For example, if the slope is \(-2\), it means that for every 1 unit increase in x, y decreases by 2 units. In simpler terms:
  • An increase in x will result in a decrease in y.
  • The steepness of the line is influenced by the absolute value of the slope.
  • The sign (negative or positive) determines the direction of the slope.
Recognizing the slope gives us insight into how variables change in relation to each other.
Positive y-intercept
The y-intercept is the point at which the line crosses the y-axis in a coordinate plane. It can be easily identified in a linear equation in the form \y = mx + b\, where \b\ is the y-intercept.

A positive y-intercept implies that the line crosses the y-axis above the origin (point \(0,0\)). This point is significant because it provides a starting value for y when x equals zero.
  • A positive y-intercept is greater than zero, indicating its position on the coordinate plane.
  • To find this value, simply look at the equation; here, it's \3\ in \y = -2x + 3\.
  • The y-intercept gives the elevation of the line when x is zero.
A clear understanding of the y-intercept helps in correctly plotting the line on a graph.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is composed of two axes:
  • The x-axis runs horizontally.
  • The y-axis runs vertically.
These axes divide the plane into four quadrants, helping us define the position of any point.
  • Coordinates are written as \(x, y\), representing horizontal and vertical placement.
  • Positive x-values move points to the right, while positive y-values move them upward.
  • The origin is where the x and y axes meet, at \(0,0\).
Knowing how to navigate the coordinate plane is crucial for graphing equations efficiently.
Plotting Points
Plotting points is a foundational task when graphing lines or curves in mathematics. It involves placing a point on the coordinate plane based on its coordinates, \(x, y\). This step is crucial for visualizing graphs.

To plot a point:
  • Start at the origin (\(0,0\)). Move horizontally according to the first number in the pair (x) along the x-axis.
  • From there, move vertically according to the second number (y) along the y-axis.
Each plotted point represents a solution to a function or equation, offering a visual understanding of its behavior.
  • Connect these points with a straight line to graph a linear equation.
  • Ensure points are accurately plotted for a precise representation of the equation.
Mastering how to plot points allows you to transform abstract equations into comprehensible visuals.

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Most popular questions from this chapter

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{5}{6 x-1}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. contains \((3,0)\) and \((7,-2)\)

The median hourly wage of an embalmer in Illinois in 2002 was \(\$ 17.82 .\) Seth's earnings, \(E\) (in dollars), for working \(t\) hr in a week can be defined by the function \(E(t)=17.82 t .\) (Source: www.igpa.uillinois.edu) a) How much does Seth earn if he works 30 hr? b) How much does Seth earn if he works 27 hr? c) How many hours would Seth have to work to make \(\$ 623.70 ?\) d) If Seth can work at most 40 hr per week, what is the domain of this function? e) Graph the function.

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(5,-1),(7,3)\\\&L_{2}:(-6,0),(4,5)\end{aligned}$$

since \(1997,\) the population of North Dakota has been decreasing by about 3290 people per year. The population was about \(650,000\) in 1997 . a) Write a linear equation to model this data. Let \(x\) represent the number of years after \(1997,\) and let \(y\) represent the population of North Dakota. b) Explain the meaning of the slope in the context of the problem. c) According to the equation, how many people lived in North Dakota in \(1999 ?\) in \(2002 ?\) d) If the current trend holds, in what year would the population be \(600,650 ?\)
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