Problem 32 - Algebra 1

Question

Graph the equation. $$ y=-\frac{1}{2} x-3 $$

Step-by-Step Solution

Answer

The graph of the equation \(y = -\frac{1}{2} x - 3\) is a straight line that passes through the points (0, -3) and (2, -4).

Step 1: Identify the Slope and y-intercept

The equation is already in the slope-intercept form \(y = mx + b\). Here, the slope 'm' is -1/2 and the y-intercept 'b' is -3.

Step 2: Plot the y-intercept

Begin by plotting the y-intercept, which is the point where the line crosses the y-axis. As the y-intercept 'b' is -3, plot a point at (0, -3) on the graph.

Step 3: Use the slope to find the next point

The slope is the ratio of vertical change (rise) over horizontal change (run). In this case, the slope 'm' is -1/2 which means for every 2 units you move to the right, you move 1 unit down. From the y-intercept (0, -3), move 2 units to the right and 1 unit down and plot the next point on the graph at (2, -4).

Step 4: Draw the Line

Now that you have two points, (0, -3) and (2, -4), you can draw a line passing through these points. This line represents the graph of the equation \(y = -\frac{1}{2} x - 3\).

Key Concepts

Understanding the Slope-Intercept Form

When graphing linear equations, one of the most intuitive and frequently used methods is the slope-intercept form, represented by the equation \(y = mx + b\). This form is incredibly useful because it clearly displays the two main characteristics of a line: its slope (\(m\)) and its y-intercept (\(b\)).

The slope measures how steep the line is, indicating whether the line is rising or falling as it moves from left to right. The y-intercept is the point where the line crosses the y-axis. This happens when the value of \(x\) is zero. In the exercise given, the equation \(y = -\frac{1}{2}x - 3\) reveals a slope of -1/2 and a y-intercept at -3.

Using this form makes plotting linear equations straightforward. Always start by identifying the slope and y-intercept from the equation. This allows you to plot the first point on the y-axis and use the slope to find subsequent points, ultimately leading to a graph representing the equation accurately and efficiently.

How to Plot the Y-Intercept

The y-intercept is the starting point for graphing a line using the slope-intercept form. It is where the line will cross the y-axis, thus the term 'y-intercept'. In our exercise, the y-intercept is -3, which is the value of \(b\) in the equation \(y = mx + b\).

To plot the y-intercept, look for the y-axis on the graph (this is the vertical axis). Since the y-axis corresponds to the value when \(x = 0\), simply find -3 on the y-axis and place a point there. This point is labeled as (0, -3). It is pivotal because it is the point through which the line will pass, serving as a reliable foundation for drawing the rest of the line as dictated by the slope.

Understanding Slope Ratio

The slope ratio is a measure of a line's steepness and direction, defined as the 'rise over run'. In simpler terms, it's how much the line goes up (or down) for a certain horizontal distance along the graph. The slope is typically represented as a fraction \(\frac{rise}{run}\).

In our example, the slope is -1/2, indicating a 'rise' of -1 (which means the line moves down) for every 'run' of 2 (moves to the right). From the y-intercept (0, -3), you would move 2 units to the right to be consistent with the ‘run’ part of our slope, then move 1 unit down due to our 'rise' being negative. You end up with the point (2, -4). This ratio is exceptionally helpful for finding additional points through which the line passes, and with at least two points, you can draw a precise line extending in both directions. Understanding the slope ratio makes graphing linear equations an orderly and exact process.