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Chapter 11: Problem 195

In the following exercises, identify the most convenient method to graph each line. $$ y=-\frac{1}{4} x+3 $$

Short Answer

Expert verified
Graph using the slope-intercept form; plot points (0, 3) and (4, 2), then draw the line through them.

Step by step solution

01

Determine the equation form

The given equation is in slope-intercept form, which is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. In this case, the equation \(y = -\frac{1}{4} x + 3\) gives the slope \(m = -\frac{1}{4}\) and the y-intercept \(b = 3\).
02

Identify the y-intercept

Identify the y-intercept \(b\). This is the point where the line crosses the y-axis. For the equation \(y = -\frac{1}{4} x + 3\), the y-intercept is 3. This provides the point \((0, 3)\) on the graph.
03

Use the slope to find another point

The slope \(m = -\frac{1}{4}\) indicates that for every 1 unit the x-coordinate increases, the y-coordinate decreases by \(\frac{1}{4}\) units. Starting from the y-intercept \((0, 3)\), move 4 units to the right (increase x by 4) and 1 unit down (decrease y by 1). This brings us to the point \((4, 2)\).
04

Plot the points and draw the line

Plot the two points \((0, 3)\) and \((4, 2)\) on a graph. Once these points are plotted, draw a straight line through them. This line represents the equation \(y = -\frac{1}{4} x + 3\).

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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 of a linear equation is one of the most convenient ways to graph a line. It is written as:
\( y = mx + b \).
Here:
  • \( m \) represents the slope of the line.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
In the provided equation \( y = -\frac{1}{4} x + 3 \):
  • The slope, \( m \), is -\( \frac{1}{4} \).
  • The y-intercept, \( b \), is 3.
Understanding this form makes it easy to quickly plot the graph of a linear equation. The y-intercept gives you a starting point on the graph, and the slope tells you how to move from one point to another.
Plotting Points
Plotting points on a graph is essential to visualizing a linear equation. With the equation in slope-intercept form, your first point comes from the y-intercept. For the equation \( y = -\frac{1}{4} x + 3 \), the y-intercept is 3. This translates to the point (0, 3) on the graph.
After identifying the y-intercept, use the slope to find additional points. The slope \( -\frac{1}{4} \) means that for every 1 unit increase in x, y decreases by \( \frac{1}{4} \) units.
  • Start from the y-intercept (0, 3).
  • Move 4 units to the right (increase x by 4).
  • Move 1 unit down (decrease y by 1).
Doing this, you arrive at the point (4, 2). Now you have two points: (0, 3) and (4, 2). Plot both of these on the graph.
Finding y-Intercept
The y-intercept is a crucial part of graphing a line because it gives you a starting point. It is where the line crosses the y-axis, which means the x-coordinate is 0. For any equation in the form \( y = mx + b \), the y-intercept is the constant term \( b \).
For the equation \( y = -\frac{1}{4} x + 3 \), the y-intercept \( b \) is 3. This point is written as (0, 3).
With this point, you can start plotting your graph. From the y-intercept, you can then use the slope to determine the direction and steepness of the line. Remember:
  • The y-intercept gives you one definite point on the graph.
  • This point helps in setting the initial position of the line.
By combining the y-intercept and the slope, you can easily draw the linear equation on the graph.

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Most popular questions from this chapter

In the following exercises, graph by plotting points. $$ y=4 x-3 $$

In the following exercises, graph each equation. $$ y=2 x $$

In the following exercises, model each slope. Draw a picture to show your results. $$ \frac{3}{2} $$

In the following exercises, find the intercepts. $$ y=\frac{3}{4} x-12 $$

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