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When graphing a linear function, starting with the equation in slope-intercept form is a great approach. The slope-intercept form of a line is given by: \[ y = mx + b \] Here, \(m\) represents the slope, while \(b\) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. In the exercise with the equation \(y = x + 2\), the slope \(m = 1\) and the y-intercept \(b = 2\). Understanding these parts allows us to easily visualize how the line behaves:

• If the slope \(m\) is positive, the line rises from left to right. When \(m = 1\), it means for every unit we move right along the x-axis, we move up one unit.
• The y-intercept \(b = 2\) tells us that the line crosses the y-axis at \(y = 2\).

This foundational understanding of the equation empowers you to draw the graph accurately.

The domain and range are critical aspects of understanding any function, providing insights into its behavior and limits. The domain refers to all the possible x-values (inputs) of a function, whereas the range covers all the possible y-values (outputs). For a linear function like \(y = x + 2\), the situation simplifies a bit. Since no restrictions are applied to \(x\), the domain is all real numbers, written as \((-\infty, \infty)\). You can plug any real number into the equation and find a corresponding y-value. Similarly, because the line extends infinitely in both vertical directions, all y-values are also possible, making the range (-∞, ∞). This information is crucial when assessing the function's behavior on a graph or within real-life problems.

An essential part of graphing the equation is plotting points, helping translate mathematical functions into visual diagrams. Let's explore this with \(y = x + 2\). Start by plotting the y-intercept directly. For this line, the y-intercept is (0, 2), so mark this point on the y-axis. This is your initial anchor. Next, use the slope \(m = 1\) to determine your next points. A slope of 1 implies that for every step right, you move one step up. This consistent pattern helps you plot successive points, such as (1, 3), (2, 4), and so forth.

• From (0, 2), go right 1 unit and up 1 unit and mark (1, 3).
• Continue this pattern to lay down more points for a more accurate line.

After plotting these points, draw a straight line through them, extending in both directions. These careful steps will ensure an accurate visual representation of the linear function.