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Chapter 3: Problem 39

Graph equation by hand. \(y=-\frac{2}{3} x+4\)

Short Answer

Expert verified
Plot the y-intercept at (0, 4), then use the slope to find a second point, and draw the line through both points.

Step by step solution

01

Identify the Slope and y-Intercept

The given equation is in slope-intercept form, which is written as \(y = mx + b\). Here, the slope \(m\) is \(-\frac{2}{3}\) and the y-intercept \(b\) is 4.
02

Plot the y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is 4, so put a point at (0, 4) on the y-axis.
03

Use the Slope to Find Another Point

The slope \(m\) is \(-\frac{2}{3}\), which means that for every 3 units you move to the right on the x-axis, you move 2 units down on the y-axis. From the point (0, 4), move 3 units to the right to (3, 4) and then 2 units down to (3, 2). Plot this point.
04

Draw the Line

With the two points plotted, draw a straight line through them to represent the equation \(y = -\frac{2}{3}x + 4\).

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Key Concepts

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

slope
The slope of a line indicates its steepness and direction. The slope is defined as the change in the y-coordinate divided by the change in the x-coordinate.
Mathematically, it is represented as \( m = \frac{\Delta y}{\Delta x} \). In the equation \( y = -\frac{2}{3}x + 4 \), the slope \( m \) is \( -\frac{2}{3} \).
This means for every 3 units you move right on the x-axis, you move 2 units down on the y-axis.
  • A positive slope indicates the line rises as you move from left to right.
  • A negative slope indicates the line falls as you move from left to right.
  • A larger slope value means a steeper line.
Understanding slope helps you graph the line accurately and understand its behavior over its domain.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. In other words, it is the value of y when x is 0.
For the given equation, \( y = b \) when \( x = 0 \), hence the y-intercept is \( b \).
In the equation \( y = -\frac{2}{3} x + 4 \), the y-intercept \( b \) is 4. This means the line will cross the y-axis at the point (0, 4).
  • The y-intercept represents the starting point to plot the graph.
  • It is crucial because it gives a fixed reference point on the graph.
Always plot the y-intercept first when graphing a linear equation.
slope-intercept form
The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
This form makes it easy to identify the slope and y-intercept directly from the equation.
In the equation \( y = -\frac{2}{3} x + 4 \), the slope-intercept form shows that the slope is \( -\frac{2}{3} \) and the y-intercept is 4.
  • The slope-intercept form provides a quick way to graph your line by identifying both the slope and y-intercept.
  • Starting with the y-intercept and using the slope, you find another point to plot the graph accurately.
Mastering the slope-intercept form is fundamental for graphing linear equations efficiently.
plotting points
Plotting points is a crucial step in graphing a linear equation.
Here is how to do it:
First, identify and plot the y-intercept. In the equation \( y = -\frac{2}{3} x + 4 \), start by plotting the y-intercept, which is the point (0, 4).
Next, use the slope to find another point. With a slope of \( -\frac{2}{3} \), from the y-intercept (0, 4), move 3 units to the right (to x=3) and 2 units down (to y=2). Plot the point (3, 2).
  • Using the slope helps consistently find points to plot the line accurately.
  • Ensuring you have at least two points plotted allows you to visualize the actual line.
Finally, draw a straight line through the plotted points to complete the graph of the equation.

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