Problem 48 - Algebra 2

Question

Graph the line with the given equation. \(y=\frac{1}{2} x\)

Step-by-Step Solution

Answer

Plot points (0, 0) and (2, 1), then draw the line connecting them.

Step 1: Understand the Equation Form

The equation of the line given is in the form of a slope-intercept equation: \(y = mx + b\). In this equation, \(m\) represents the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. In the equation \(y = \frac{1}{2}x\), the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is 0 because there is no constant term added to this equation.

Step 2: Identify Key Points

To graph the line, we need at least two points. The simplest point to start with is the y-intercept, which is (0, 0) for this line. Using the slope \(\frac{1}{2}\), which means "rise over run," we can determine the second point. The slope \(\frac{1}{2}\) tells us that for every 1 unit we move up (rise), we move 2 units to the right (run). Starting from (0, 0), moving 2 units to the right and 1 unit up, we reach the point (2, 1).

Step 3: Plot the Points and Draw the Line

On a coordinate plane, plot the points (0, 0) and (2, 1). Connect these points using a straight line. This line represents the graph of the equation \(y = \frac{1}{2}x\). Ensure the line extends in both directions to cover the entire graph, as it continues infinitely. Make sure that the line is straight and reflects the constant slope across the xy-plane.

Key Concepts

Slope-Intercept Form

One of the most straightforward ways to represent a linear equation is through the slope-intercept form, which is expressed as \( y = mx + b \). In this equation:

• \( m \) denotes the slope, which is the measure of how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

This form is especially useful for graphing because it immediately tells us two vital pieces of information about the line: its slope and where it intersects the y-axis. For instance, in the equation \( y = \frac{1}{2}x \), you can identify:

• The slope \( m \) is \( \frac{1}{2} \), indicating a gradual incline.
• The y-intercept \( b \) is 0, meaning the line passes through the origin \((0, 0)\).

With just these components, you can sketch the graph by starting at the y-intercept and using the slope to determine additional points.

Coordinate Plane

A coordinate plane is a two-dimensional grid that helps us visualize and work with equations, especially linear ones. It consists of two axes: the horizontal x-axis and the vertical y-axis, which intersect at a point called the origin \((0, 0)\).
Here's how you can use a coordinate plane effectively:

• Each point on the plane is represented by a pair of numbers \((x, y)\), known as coordinates.
• These coordinates indicate a specific location, with \( x \) showing the position on the x-axis and \( y \) showing the position on the y-axis.

To graph the equation \( y = \frac{1}{2}x \), start by plotting the y-intercept on the coordinate plane. This provides a reference point from which you can plot additional points using the slope. Grasping how the coordinate plane functions is key to successfully graphing linear equations, as it allows you to interpret and represent mathematical relationships visually.

Y-Intercept

The y-intercept is a core concept in graphing linear equations. It tells us where the line crosses the y-axis. When the equation of a line is given in slope-intercept form \( y = mx + b \), the y-intercept is the constant term \( b \).
Understanding the y-intercept is simple yet powerful because:

• The y-intercept can be directly observed at point \((0, b)\) on the graph.
• It serves as the starting point for graphing the line since no x-axis movement is required to plot it.

In our example, the equation \( y = \frac{1}{2}x \) has a y-intercept of 0, representing the origin \((0, 0)\). By identifying this point, you can track the line's path using the slope's behavior from that point onward. Comprehending the y-intercept's role enables better understanding of how changes in the equation affect the entire graph.