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Graphing linear functions involves plotting points on a coordinate plane and drawing a straight line through these points. Linear functions are usually in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

Start by identifying the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis. Then, use the slope to determine the direction and steepness of the line.

To graph \( g(x) = -x + 3 \), first plot the \( y \)-intercept (0,3). From there, use the slope to find another point on the line. Since the slope is -1, move down 1 unit and right 1 unit from the \( y \)-intercept to plot the next point.

With at least two points on your graph, draw a straight line connecting these points. This line represents the function \( g(x) = -x + 3 \).

The slope of a linear function indicates its steepness and direction. It's the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. Denoted by \( m \), the slope can be positive, negative, zero, or undefined.

In the function \( g(x) = -x + 3 \), the coefficient of \( x \) is -1. This means the slope \( m \) is -1.

A negative slope means the line decreases from left to right. For every unit you move to the right, the line goes down by the slope's magnitude. In this case, for each step right, go 1 step down.

Remember, the slope is crucial for graphing because it tells us how to move from the \( y \)-intercept to find additional points on the graph.

The \( y \)-intercept is where the line crosses the \( y \)-axis. It's represented by the constant term \( b \) in the slope-intercept form \( y = mx + b \).

For the function \( g(x) = -x + 3 \), the \( y \)-intercept \( b \) is 3. This means the line crosses the \( y \)-axis at the point (0,3).

When graphing the function, start by plotting this point. The \( y \)-intercept provides a starting point for drawing the line. From there, use the slope to find additional points.

The \( y \)-intercept is a quick way to visualize where a line starts on the graph, making it easier to draw accurate linear functions.