Question

Find the slope and y-intercept of each line. Graph the line. $$ y=\frac{1}{2} x+2 $$

Short answer

Slope: \(\frac{1}{2}\), y-intercept: 2.

Step-by-step solution

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 \(\frac{1}{2}\).

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.

Plot the y-intercept

To graph the line, start by plotting the y-intercept (0, 2) on the coordinate plane.

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.

Draw the line

Draw a straight line through the points (0, 2) and (2, 3) to complete the graph of the line.

Key concepts

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.

Understanding the slope helps in predicting the behavior of the line on the graph.

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.

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.

This line represents all solutions to the equation \( y = \frac{1}{2} x + 2 \). Each point on the line satisfies the equation. By following this method, you can graph any linear equation in slope-intercept form.

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):

• x is the distance from the y-axis.
• y is the distance from the x-axis.

When graphing linear equations:

• The y-intercept is where the line crosses the y-axis.
• The slope determines the line's steepness and direction.

The coordinate plane allows us to visually represent equations and better understand their relationships. By plotting points and drawing lines, we can see how the equations behave in space.