91Ó°ÊÓ

Linear equations describe a straight line in the coordinate plane. The slope is one of the most important features. The slope indicates how steep the line is. In a linear equation of the form \( y = mx + b \), the term 'm' represents the slope. It tells you how much \(y\) changes for a unit increase in \(x\). A larger slope means a steeper line.
In the exercise, we have \( f(x) = 4x - \frac{1}{4} \). To find the slope, we need to identify 'm'. If you compare this equation to the general form \(y = mx + b\), you can see that \(m = 4\). Thus, the slope is 4.
A slope of 4 means that for every 1 unit you move to the right on the x-axis, the y-value goes up by 4 units. So, the line is quite steep, as it rises quickly.

Graphing linear functions involves plotting points and drawing a line through them. The general form of a linear equation is \(y = mx + b\). In this form, 'm' represents the slope and 'b' represents the y-intercept. To graph, start by plotting the y-intercept, then use the slope to find other points.
Take the function \(f(x) = 4x - \frac{1}{4}\) from our exercise.
- First, plot the y-intercept, which is \(-\frac{1}{4}\). This point is where the line crosses the y-axis.
- Next, use the slope. With a slope of 4, for every 1 unit you move right along the x-axis, move 4 units up along the y-axis.
Repeat this process to plot a few points, then draw a line through them.
This will give you the graph of the function.

The y-intercept of a linear equation is the point where the line crosses the y-axis. This occurs when \(x = 0\). It is represented by 'b' in the general form \(y = mx + b\). Finding the y-intercept is simple - just look at the constant term in the equation.
In the exercise, the equation is \(f(x) = 4x - \frac{1}{4}\). Comparing this to the general form, we see that \(b = -\frac{1}{4}\). Therefore, the y-intercept is \(-\frac{1}{4}\).
Graphically, this means the line crosses the y-axis at the point \((0, -\frac{1}{4})\). This point is essential for accurately graphing the function and understanding where it starts on the y-axis.