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Chapter 13: Problem 4

On the same axes, graph \(y=-\frac{2}{3} x+b\) for \(b=0,3,6,-3,\) and \(-6\).

Short Answer

Expert verified
The graphs for the given equation with different values of \(b\) all have the same slope of \(-\frac{2}{3}\). They start from different points along the y-axis (0, 3, 6, -3, -6) based on the value of \(b\). These points are the y-intercept points for each line.

Step by step solution

01

Identifying the slope and y-intercept

The equation of the line is given in the slope-intercept form \(y=mx+b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. Here, the slope \(m = -\frac{2}{3}\) and the y-intercept \(b\) takes 5 different values: 0, 3, 6, -3, -6.
02

Graphing for \(b=0\)

The line passes through the origin (0, 0) because \(b=0\) for this case. Due to the slope of \(-\frac{2}{3}\), for every 3 units increase in x, y decreases by 2 units.
03

Graphing for \(b=3\)

The y-intercept for this case is (0, 3). Starting from this point, the line is plotted considering the negative slope of \(-\frac{2}{3}\). This means for every 3 units increase in x, y decreases by 2 units.
04

Graphing for \(b=6\)

Starting from the y-intercept (0, 6), the line is plotted maintaining the negative slope of \(-\frac{2}{3}\). For every 3 units increase in x, y decreases by 2 units.
05

Graphing for \(b=-3\)

The y-intercept for this case is (0, -3). Starting from this point, the line follows the negative slope of \(-\frac{2}{3}\). Therefore for every 3 units increase in x, y decreases by 2 units.
06

Graphing for \(b=-6\)

Starting from the y-intercept (0, -6), the line is plotted by maintaining the slope of \(-\frac{2}{3}\). This means for every 3 units increase in x, y decreases by 2 units.

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Key Concepts

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

Slope-Intercept Form
When graphing linear equations, the slope-intercept form is particularly useful. It is expressed as \( y = mx + b \), where:
  • \( m \): Represents the slope of the line. This is how steep the line is and in which direction it slants.
  • \( b \): This is where the line crosses the y-axis, known as the y-intercept.
Using this form, you can quickly identify the key features of a line. Once you know \( m \) and \( b \), you can easily draw the line by:
  • Plotting the y-intercept (starting point on the y-axis).
  • Using the slope to determine the tilt of the line by counting the rise over run.
This method helps in understanding and graphing the line with minimal calculations.
Practicing with different values of \( b \) and \( m \) will make your understanding even better.
Y-Intercept
The y-intercept is a crucial part of understanding where a line is situated on a graph. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented by \( b \).
This number tells you the exact point where the line crosses the y-axis, which is the vertical line on the graph.
  • For example, if \( b = 3 \), the line crosses the y-axis at the point (0, 3).
  • If \( b = -6 \), it crosses at (0, -6).
      • No matter what the slope is, the position of this intercept does not change unless \( b \) changes.
      This makes graphing multiple lines on the same axis, like in the original exercise, clearer and more organized. By adjusting \( b \), you can model different scenarios and see immediate changes in your graph.
Negative Slope
A negative slope is an interesting feature of a line that indicates the direction it moves in. A negative value in the slope \( m \) means that as you move along the x-axis to the right, the line will move downwards.
This shows a decrease in the y-value as x increases.
  • The specific negative slope in the exercise is \( -\frac{2}{3} \).
  • This means for every 3 units you move to the right on the x-axis, you move 2 units downwards on the y-axis.
Working with negative slopes can initially be confusing, but it's helpful to remember the line 'descends' or goes down moving from left to right on a graph.
It's the opposite of a positive slope, where a line would rise.
  • If you graph a line with a negative slope, it will appear to lean backward, much like a slide.
Understanding which way your line will go with different slopes helps in predicting the graph's behavior without even drawing it completely.

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Most popular questions from this chapter

Graph the points \(A(-5,0), B(3,2), C(5,6),\) and \(D(-3,4) .\) Then show that \(A B C D\) is a parallelogram by two different methods. a. Show that one pair of opposite sides are both congruent and parallel. b. Show that the diagonals bisect each other (have the same midpoint).

horizontal line through \((3,1)\)

Find and then compare lengths of segments. Quadrilateral \(T A U L\) has vertices \(T(4,6), A(6,-4), U(-4,-2),\) and \(L(-2,4) .\) Show that the diagonals are congruent.

(a) find the lengths of the sides of \(\triangle R S T,\) (b) use the converse of the Pythagorean Theorem to show that \(\triangle R S T\) is a right triangle, and (c) find the product of the slopes of \(\overline{R T}\) and \(\overline{S T}\). \(R(4,3), S(-3,6), T(2,1)\)

Find each vector sum. Then illustrate each sum with a diagram like that on page 541. $$(-8,2)+(4,6)$$
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