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Chapter 11: Problem 60

In the following exercises, graph by plotting points. $$ y=-\frac{4}{5} x-1 $$

Short Answer

Expert verified
Plot the y-intercept at \((0, -1)\), then use the slope \(-\frac{4}{5}\) to find more points like \((5, -5)\). Draw the line through the points.

Step by step solution

01

Identify the equation form

The given equation is in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the given equation, \(y = -\frac{4}{5}x - 1\), the slope \(m\) is \(-\frac{4}{5}\) and the y-intercept \(b\) is \(-1\).
02

Plot the y-intercept

Start by plotting the y-intercept. In this case, the y-intercept is \(-1\). Place a point on the graph at \( (0, -1) \).
03

Use the slope to find another point

The slope \( m \) is \(-\frac{4}{5}\). This means for every 5 units you move to the right (positive x-direction), you move down 4 units (negative y-direction). Starting from the y-intercept \((0, -1)\), move 5 units to the right to \( x = 5 \) and 4 units down to \( y = -5 \). Place a point at \( (5, -5) \).
04

Plot additional points if needed

You can plot more points following the slope rule to ensure accuracy. For instance, start from the y-intercept \((0, -1)\), move 5 units to the left \(x = -5\) and 4 units up \(y = 3\). Place another point at \((-5, 3)\).
05

Draw the line

Once enough points are plotted, draw a straight line through all the points. This line represents the graph of the equation \( y = -\frac{4}{5} x - 1 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form is commonly used to represent linear equations. It's written as
\[ y = mx + b \].
Here, m denotes the slope of the line, and b stands for the y-intercept鈥攖he point where the line crosses the y-axis. For example, in the equation \[ y = -\frac{4}{5}x - 1 \], the slope m is \( -\frac{4}{5} \) and the y-intercept b is -1. This form makes it easy to graph the line since you can immediately identify the slope and the starting point on the y-axis.
plotting points
Plotting points is essential for graphing linear equations. Start by identifying points from the equation. Begin with the y-intercept, then use the slope to locate more points. In the case of \( y = -\frac{4}{5}x - 1 \):
  • First, plot the y-intercept (0, -1).
  • Use the slope \( -\frac{4}{5} \): move 5 units to the right and 4 units down to find the next point, (5, -5).
  • Find an additional point by moving 5 units to the left and 4 units up, resulting in (-5, 3).

These points help to accurately draw the line representing the equation.
linear equations
Linear equations graph as straight lines. They often appear in the format \( y = mx + b \). This form indicates a relationship between x and y that maintains constant rates of change. For example, in \( y = -\frac{4}{5}x - 1 \), every unit change in x causes a change proportional to the slope m in y. This straight-line relationship makes it easy to visualize and predict values along the graph.
y-intercept
The y-intercept is where the line crosses the y-axis. It鈥檚 the point where x equals zero. In the equation \( y = -\frac{4}{5}x - 1 \), the y-intercept b is -1. To plot it:
  • Locate the y-axis point at (0, -1).
  • Mark this point as the starting position for the line.

Identifying the y-intercept is the initial step in graphing linear equations.
slope
The slope of a line represents its steepness and direction. It's denoted as m in the equation \( y = mx + b \). The slope is calculated as the ratio of vertical change (rise) to horizontal change (run). In \( y = -\frac{4}{5}x - 1 \), the slope m is \( -\frac{4}{5} \), meaning:
  • For a positive movement of 5 units along the x-axis, the y-axis value decreases by 4 units.
  • This negative slope indicates a downward trend as you move from left to right.

Understanding the slope helps you predict and plot additional points on the graph accurately.

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Most popular questions from this chapter

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (2,1),(4,5) $$

In the following exercises, graph each pair of equations in the same rectangular coordinate system. $$ y=5 x \text { and } y=5 $$

In the following exercises, identify the most convenient method to graph each line. $$ y=\frac{2}{3} x-1 $$

In the following exercises, graph the line given a point and the slope. $$ (-3,3) ; m=2 $$

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (3,5),(4,-1) $$
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