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Chapter 4: Problem 266

Graph each line with the given point and slope. (1,4)\(; m=-\frac{1}{2}\)

Short Answer

Expert verified
The equation of the line is \( y = -\frac{1}{2}x + 4.5 \).

Step by step solution

01

Understand the Slope-Intercept Form

The equation of a line in slope-intercept form is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify Given Information

We are given a point (1,4) and a slope \( m = -\frac{1}{2} \). We need to use this information to find the y-intercept \( b \).
03

Plug in the Point into the Slope-Intercept Equation

Substitute the point (1,4) into the slope-intercept form equation along with the slope. This gives us the equation: \[ 4 = -\frac{1}{2}(1) + b \]
04

Solve for the Y-Intercept

Solve the equation to find the y-intercept \( b \). \[ 4 = -\frac{1}{2} + b \] Add \( \frac{1}{2} \) to both sides: \[ 4 + \frac{1}{2} = b \] \[ b = 4.5 \] or \[ b = \frac{9}{2} \]
05

Write the Equation of the Line

Now, substitute \( m \) and \( b \) back into the slope-intercept equation: \[ y = -\frac{1}{2}x + 4.5 \] or \[ y = -\frac{1}{2}x + \frac{9}{2} \]
06

Graph the Line

To graph the line, start at the y-intercept \( (0, 4.5) \). From this point, use the slope \( -\frac{1}{2} \). This means for every 2 units you move to the right, you move 1 unit down. Plot several points this way and draw the line through those points.

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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 a way to express the equation of a straight line. The general form is \(
y = mx + b \), where \( m \) represents the slope and \( b \) signifies the y-intercept.
This form is valuable because it directly shows how the line behaves.
The slope \( m \) tells us how steep the line is, and the y-intercept \( b \) tells us where the line crosses the y-axis.
By simply looking at an equation in the slope-intercept form, we can understand the general shape and position of the line.
This form simplifies graphing because you can identify these key components easily.
y-intercept
The y-intercept \( b \) is the point where the line crosses the y-axis.
In the equation \(
y = mx + b \), the y-intercept is represented by the value of \( b \.\) For example, in the equation \( y = -\frac \{ 1 \}\backslash\backslash 2 \} x + 4.5 \), the y-intercept is 4.5.
This means the line will pass through the point \( (0, 4.5) \) on the coordinate plane.
Finding the y-intercept involves setting \( x=0 \) in the slope-intercept equation and solving for \( y \).
plotting points
Plotting points is essential for graphing a line.
First, you start with a known point, like the y-intercept \( (0, 4.5) \), and then use the slope to find other points on the line.
For instance, with a slope of \(-\frac \{ 1 \}\backslash\backslash 2 \), every time you move 2 units to the right on the x-axis, you'll move down 1 unit on the y-axis.
Plot these points and connect them to form the linear graph.
This technique helps visualize how the equation relates to the graph.
slope
The slope \( m \) of a line measures its steepness.
It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
A negative slope means the line slopes downward from left to right, while a positive slope means it slopes upward.
For example, a slope of \(-\frac \{1 \}\backslash\backslash2 \) means for every 2 units you move to the right, you'll move 1 unit down.
This is crucial for understanding the direction and angle of the line on the graph.

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