Question

Find the slope and \(y\) intercept of each straight line and make a graph. $$y=-\frac{1}{2} x-\frac{1}{4}$$

Short answer

The slope of the line is \(-\frac{1}{2}\) and the y-intercept is \(-\frac{1}{4}\). The line can be graphed using these values.

Step-by-step solution

Identify the Slope

The equation of the line is given in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, the coefficient of \(x\) is \(-\frac{1}{2}\), which means the slope \(m = -\frac{1}{2}\).

Identify the Y-Intercept

In the slope-intercept form of the equation, the constant term represents the y-intercept. In this case, the y-intercept \(b\) is \(-\frac{1}{4}\).

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. On the y-axis, locate the point \((0, -\frac{1}{4})\) and mark it as the starting point of the line.

Use the Slope to Find Another Point

The slope \(-\frac{1}{2}\) means that for every unit you move to the right along the x-axis, you move down by half a unit on the y-axis. From the y-intercept \((0, -\frac{1}{4})\), move 1 unit to the right to \((1, -\frac{1}{4})\) and then move down \(\frac{1}{2}\) unit to find the next point \((1, -\frac{1}{4} - \frac{1}{2})\) which simplifies to \((1, -\frac{3}{4})\).

Draw the Line

With the two points \((0, -\frac{1}{4})\) and \((1, -\frac{3}{4})\), draw a straight line through these points to graph the equation \(y = -\frac{1}{2}x - \frac{1}{4}\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing linear relationships. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

For the given equation, \( y = -\frac{1}{2}x - \frac{1}{4} \), it's clear that the slope-intercept form has been provided. The value in front of the \( x \), which is \( -\frac{1}{2} \), is the slope. This tells us how steep the line is and in which direction it tilts. Since the slope is negative, we know our line is going to slope downwards as we move from left to right.

Moreover, the slope can tell us how to move from one point on the line to the next. In this case, a slope of \( -\frac{1}{2} \) means that for every step right (positive direction along the x-axis), we take a half step down (negative direction along the y-axis). This creates a consistent decline across the graph as we plot more points based on our slope.

Y-Intercept

The y-intercept is perhaps one of the most straightforward concepts in graphing linear equations. It's the point where the line crosses the y-axis and is represented by the \( b \) in the slope-intercept equation \( y = mx + b \).

For the equation in question, the y-intercept is \( -\frac{1}{4} \). This means that when \( x \) is zero, \( y \) will equal \( -\frac{1}{4} \), placing our y-intercept at point \( (0, -\frac{1}{4}) \) on the graph.

When graphing, you should always start by plotting the y-intercept first since it's the anchor point for your line. It's the 'starting block' from which you will use the slope to find other points. A common mistake is to confuse the y-intercept with any point that has a 'zero' in it, but remember, it's specifically the point where the line touches the y-axis.

Slope of a Line

The slope is a measure of how steep a line is and the direction in which it tilts. Defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line, the slope is often represented by the letter \( m \).

In our example with the equation \( y = -\frac{1}{2}x - \frac{1}{4} \), the slope is \( -\frac{1}{2} \). This negative value indicates a downward slope from left to right. To find another point using the slope, we follow the 'rise over run' concept. Since our slope is negative, we'd rise (move up) or fall (move down) \( \frac{1}{2} \) on the y-axis for every unit we run (move right) on the x-axis.

A trick to better understanding slope is to rewrite it as a fraction; \( -\frac{1}{2} \) can be seen as \( -\frac{1}{2} \) over 1. It provides a clear view of how to move across the graph: down 1 unit for every 2 units to the right. Keeping this consistent, plotting additional points becomes easier, which then allows you to draw the line more accurately on the graph.