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Chapter 3: Problem 23

In the following exercises, graph by plotting points. $$ y=-\frac{3}{2} x+2 $$

Short Answer

Expert verified
Plot the points (0, 2) and (2, -1) then draw the line passing through them.

Step by step solution

01

- Understand the Equation

The given equation is in slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(m = -\frac{3}{2}\) and \(b = 2\).
02

- Identify the Y-Intercept

The y-intercept is the value of \(y\) when \(x = 0\). In this equation, \(b = 2\), so the y-intercept is at the point (0, 2).
03

- Use the Slope to Find Another Point

The slope \(-\frac{3}{2}\) means that for every 2 units the graph moves to the right (positive direction along the x-axis), it moves down by 3 units (negative direction along the y-axis). Starting from (0, 2), move 2 units to the right to reach \(x = 2\), then move down 3 units to \(y = -1\). This gives another point (2, -1).
04

- Plot the Points on a Graph

Plot the points (0, 2) and (2, -1) on a Cartesian plane.
05

- Draw the Line

Connect the plotted points with a straight line. The line extends infinitely in both directions, representing all the solutions to the equation \(y = -\frac{3}{2} x + 2\).

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Key Concepts

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

Slope-Intercept Form
Understanding the equation of a line can be simplified using the slope-intercept form. This form is given by the equation: \( y = mx + b \), where
  • \( y \): the dependent variable
  • \( m \): the slope of the line
  • \( x \): the independent variable
  • \( b \): the y-intercept
The slope, \( m \), determines how steep the line is, and the y-intercept, \( b \), tells us where the line crosses the y-axis. To graph a line using the slope-intercept form, it鈥檚 essential to find these two parameters first. This will be your starting point and guide to plotting the entire line.
Y-Intercept
The y-intercept is a main concept in graphing linear equations. It鈥檚 where the line crosses the y-axis, giving you a starting point on the graph. In the equation \( y = -\frac{3}{2} x + 2 \), the y-intercept \( b \) is 2. This means that when \( x = 0 \), \( y \) is 2.

To summarize:
  • Locate the y-axis on your Cartesian plane.
  • Find the value of \( b \).
  • Plot the point (0, b) on the y-axis.
So, for our example, you would plot the point (0, 2) on the y-axis.
Plotting Points
Plotting points is about finding specific coordinates that satisfy the linear equation. After identifying the y-intercept, you use the slope to find another point. The slope \( -\frac{3}{2} \) means that for every 2 units you move to the right on the x-axis, you move down 3 units on the y-axis.

Steps are:
  • Start at the y-intercept point (0, 2).
  • Move right along the x-axis by 2 units.
  • Move down 3 units.
  • Plot the new point, (2, -1).
This point helps you draw a clearer, more accurate line on your graph.
Cartesian Plane
A Cartesian plane is essential for graphing any linear equation. It鈥檚 made up of two axes: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is called the origin, marked as (0, 0).

To use it:
  • Draw the x and y axes on your graph paper.
  • Label each axis with numbers to represent the scale.
  • The point (x, y) shows an exact location on the plane鈥攛 is the horizontal value, y is the vertical.
  • Start at (0, 0), and follow the x-coordinate moves right/left, and the y-coordinate moves up/down.
Plotting points like (0, 2) and (2, -1) helps visualize the linear relationship described by the equation.

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Most popular questions from this chapter

Graph the linear inequality: \(x-y<-3\)

In the following exercises, determine whether each ordered pair is a solution to the given inequality. Determine whether each ordered pair is a solution to the inequality \(y<-2 x+5:\) (a) (-3,0) (b) (1,6) (c) (-6,-2) (d) (0,1) (c) (5,-4)

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=-3 x+3 $$

In the following exercises, use the set of ordered pairs to ( ) determine whether the relation is a function, (b) find the domain of the relation, and \(\odot\) find the range of the relation. \([(-3,9),(-2,4),(-1,1),\) (0,0),(1,1),(2,4),(3,9)\\}

In the following exercises, evaluate the function: (@) \(f(2)\) (b) \(f(-1)\) (c) \(f(a) .\) $$ f(x)=2 x^{2}-x+3 $$
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