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Chapter 6: Problem 113

Graph: \(y=-\frac{2}{3} x+1 .\) (Section 3.4, Example 3)

Short Answer

Expert verified
The graph of the line \(y=-\frac{2}{3}x + 1\) has a slope of \(-\frac{2}{3}\) and y-intercept at \(1\). Starting from the y-intercept, one can find additional points on the line using the slope and then connect the points to form the line.

Step by step solution

01

Identify the Slope and the Y-intercept

From the given equation \(y=-\frac{2}{3} x+1\), the coefficient of \(x\) is the slope \(m\), and the constant term is the y-intercept \(c\). Hence, the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(c\) is \(1\).
02

Plot the Y-Intercept

Start by plotting the y-intercept on the y-axis. In this case, the y-intercept is \(1\), so put a point at \(1\) on the y-axis.
03

Use the Slope to Find the Next Point

The slope is \(-\frac{2}{3}\), which means for every \(3\) units we move to the right on the x-axis, we need to move \(2\) units down on the y-axis. Starting from the y-intercept point, move \(3\) units to the right and \(2\) units down to find the next point on the line.
04

Draw the Line

Connect the y-intercept and the point determined by the slope with a straight line. This is the graph of the line \(y=-\frac{2}{3}x + 1\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a straight line. It is written as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept. This form is incredibly useful because it allows you to easily see how the line behaves without needing to manipulate complex equations. As soon as you know the slope and the y-intercept, you can graph the line on the coordinate plane. The slope-intercept form simplifies the process to two straightforward steps: plot the y-intercept and then use the slope to find another point on the line.
Slope
Slope is a key concept when dealing with linear equations. It tells us how steep the line is and in which direction it slants. The slope is the ratio of the vertical change to the horizontal change between two points on a line. If you are given the slope as \( -\frac{2}{3} \), this means:
  • For every 3 units you move to the right, you move 2 units down.
  • A negative slope indicates that the line is falling as it moves left to right.
Slope is often denoted by the letter \( m \) in the slope-intercept form of an equation. Remember, the steeper the line, the larger the absolute value of the slope will be.
Y-Intercept
The y-intercept is the point where your line crosses the y-axis on a graph. It represents the value of \( y \) when \( x = 0 \). In the equation \( y = -\frac{2}{3}x + 1 \), the y-intercept is \( 1 \). To find this on a graph:
  • Look at the constant term in the equation, which is the value of the y-intercept.
  • Plot this point on the y-axis. For instance, place your point at \((0, 1)\).
Finding the y-intercept first helps to anchor your graph, making it easier to accurately draw the rest of the line.
Coordinate Plane
The coordinate plane is the surface on which you graph your equations. It is divided into four quadrants and consists of a horizontal axis (x-axis) and a vertical axis (y-axis). Understanding how to navigate the coordinate plane is crucial for graphing lines:
  • The point where the axes intersect is called the origin, labeled \((0, 0)\).
  • The x-axis allows you to position points horizontally, while the y-axis allows for vertical positioning.
To plot the equation \( y = -\frac{2}{3}x + 1 \), you start at the y-intercept \((0, 1)\) and use the slope to identify where the next point lies on the plane. This framework makes graphing linear equations visually intuitive and straightforward.

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Most popular questions from this chapter

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