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Magazine Chapter 2: Problem 83
Find the slope and y-intercept of each line. Graph the line. $$ y=\frac{1}{2} x+2 $$
Short Answer
Expert verified
Slope: \(\frac{1}{2}\), y-intercept: 2.
Step by step solution
01
Identify the slope
The given equation is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) represents the slope and \(b\) represents the y-intercept. By comparing, we see that \(m = \frac{1}{2}\). Therefore, the slope of the line is \(\frac{1}{2}\).
02
Identify the y-intercept
From the slope-intercept form \(y = mx + b\), the \(b\) term represents the y-intercept. In the given equation, \(b = 2\). Therefore, the y-intercept of the line is 2.
03
Plot the y-intercept
To graph the line, start by plotting the y-intercept (0, 2) on the coordinate plane.
04
Use the slope to find another point
The slope \(\frac{1}{2}\) means that for every 2 units you move horizontally to the right, you move 1 unit vertically up. Starting from the y-intercept (0, 2), move 2 units to the right to (2, 2) and then 1 unit up to (2, 3). Plot this point.
05
Draw the line
Draw a straight line through the points (0, 2) and (2, 3) to complete the graph of the line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope of a line
The slope of a line is a measure of its steepness. It tells us how much y changes for a given change in x. In the equation \( y = mx + b \), the slope is represented by \( m \). To find the slope from the equation \( y = \frac{1}{2} x + 2 \), we identify \( m = \frac{1}{2} \). This means that for every increase of 2 units in x, y increases by 1 unit. The slope indicates the direction of the line:
- If \( m > 0 \), the line rises as it moves from left to right.
- If \( m < 0 \), the line falls as it moves from left to right.
- If \( m = 0 \), the line is horizontal.
- If the slope is undefined (i.e., division by zero), the line is vertical.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when x is zero. In the equation \( y = mx + b \), the y-intercept is represented by \( b \). For the equation \( y = \frac{1}{2}x + 2 \), the y-intercept is \( b = 2 \). This means the line crosses the y-axis at (0, 2).
The y-intercept is crucial for graphing because it gives us a starting point. By plotting this point first, we can use the slope to determine the direction and steepness of the line. For instance, with an intercept at (0, 2) and a slope of \( \frac{1}{2} \), we move up 1 unit for every 2 units we move to the right.
The y-intercept is crucial for graphing because it gives us a starting point. By plotting this point first, we can use the slope to determine the direction and steepness of the line. For instance, with an intercept at (0, 2) and a slope of \( \frac{1}{2} \), we move up 1 unit for every 2 units we move to the right.
graphing linear equations
Graphing a linear equation involves plotting points and drawing a line through them. Let's graph \( y = \frac{1}{2}x + 2 \):
- First, plot the y-intercept (0, 2).
- Next, use the slope \( \frac{1}{2} \) to find additional points. From (0, 2), move 2 units to the right to (2, 2), and 1 unit up to (2, 3). Plot this point as well.
- Finally, draw a straight line through both points.
coordinate plane
The coordinate plane is a two-dimensional surface where we graph points, lines, and curves. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at the origin (0, 0).
Each point on the plane is represented by an ordered pair (x, y):
Each point on the plane is represented by an ordered pair (x, y):
- x is the distance from the y-axis.
- y is the distance from the x-axis.
- The y-intercept is where the line crosses the y-axis.
- The slope determines the line's steepness and direction.
