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Chapter 12: Problem 15

Graph. (GRAPH CANNOT COPY) $$y=-\frac{2}{5} x+1$$

Short Answer

Expert verified
The slope of the line is -2/5 and the y-intercept is 1. After plotting these on the graph, a line that passes through these points is drawn giving the graph of the equation.

Step by step solution

01

Identify the slope and y-intercept

The equation is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Here, the slope \(m = -\frac{2}{5}\) and the y-intercept \(c = 1\).
02

Plot the y-intercept

Start by placing a dot at the y-intercept, which is at the point (0, 1) on the graph.
03

Use the slope to find another point

Slope is the rise over run. In this case, the slope is -2/5, which means for every 5 units you move to the right (run), you move down 2 units (rise). From the y-intercept, move 5 units to the right and 2 units down and place a dot at this location.
04

Draw the line

Draw a straight line that passes through the two points just plotted. This line is the graph of the equation \(y = -\frac{2}{5} x + 1\).

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Key Concepts

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

Slope-Intercept Form
Linear equations can often be represented in a specific format known as the slope-intercept form. This form is expressed as \( y = mx + c \). Here, \( m \) denotes the slope of the line, and \( c \) stands for the y-intercept. It is a very convenient way to represent a line because it easily provides two key characteristics of the line: its steepness and its position.
  • The slope \( (m) \) gives information about how steep the line is and in which direction it tilts.
  • The y-intercept \( (c) \) shows where the line crosses the y-axis.
By understanding these two parts, you can quickly sketch a simple graph without needing to solve additional equations. This is why the slope-intercept form is a favorite among many when initially learning to graph linear equations.
Slope
Slope is a crucial part of linear equations as it indicates how steep a line is. It is defined as the "rise over run," meaning how much you move up or down for each step you move to the right. In our example \(y = -\frac{2}{5}x + 1\):
  • The slope \(-\frac{2}{5}\) tells us that for every 5 units we go to the right, the line moves 2 units down.
  • This indicates a downward tilt since the slope is negative.
The larger the absolute value of the slope, the steeper the line. A positive slope means the line moves upward as it goes to the right, whereas a negative slope, like in our example, means the line slopes downward.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \(y = -\frac{2}{5}x + 1\), the y-intercept is given by the constant term \(1\). This means:
  • On the graph, the line will intersect the y-axis at point \((0,1)\).
  • The y-intercept helps in determining one fixed point on the graph right at the start.
Plotting this point is typically the first step when graphing a line, as it provides a solid reference from which you can determine the other points based on the slope. It simplifies the process of graphing by giving you a vital point without any calculation.
Plotting Points
After determining the slope and y-intercept, plotting points becomes straightforward. Here's how to do it step-by-step for the equation \(y = -\frac{2}{5}x + 1\):
First, plot the y-intercept \((0,1)\) on the graph since we know the line crosses the y-axis there. Next, use the slope to find another point:
  • With the slope \(-\frac{2}{5}\), start at \((0,1)\), and move 5 units right horizontally.
  • From there, move 2 units down vertically since the rise is negative, placing you at the new point. Plot this point on the graph.
Once these two points are marked, drawing a straight line through them will give you the graph of the equation. This technique ensures accuracy and visually represents the relationship indicated by the equation.

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Most popular questions from this chapter

Does \(|y|=x-1,\) where \(x \in\\{1,2,3,4\\},\) define \(y\) as a function of \(x ?\)

If two lines have the same slope and the same \(y\) -intercept, must the graphs of the lines be the same? If not, give an example.

What effect does increasing the constant term have on the graph of \(y=m x+b ?\)

Find the slope and \(y\) -intercept of the graph of the equation. $$y=-x+7$$

Do the graphs of all straight lines have a \(y\) -intercept? If not, give an example of one that does not.
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