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Graphing linear functions is a fundamental skill in algebra that involves plotting points on a graph to form a straight line. The equation given, \(y = 3x + 3\), is in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This equation tells us that the line has a slope of 3, meaning it rises three units for every one unit it moves to the right. Additionally, the y-intercept is 3, indicating that the line crosses the y-axis at \(y = 3\).

To graph a linear equation like this one, you simply need the slope and one point, typically the y-intercept. However, ensuring accuracy can be easier with a table of values and plotting multiple points. Once the points are plotted, draw a straight line connecting them. This line represents all solutions to the equation.

The x-intercept is a specific point on a graph where the line crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept of the equation \(y = 3x + 3\), you substitute \(y = 0\) into the equation and solve for \(x\).

In this case:

• Set \(y = 0\) in the equation: \(0 = 3x + 3\).
• Subtract 3 from both sides: \(-3 = 3x\).
• Divide by 3: \(x = -1\).

Thus, the x-intercept is at the point \((-1, 0)\). This means that the graph will cross the x-axis at \(x = -1\). Recognizing the x-intercept provides valuable insight into where the line will interact with the x-axis.

The y-intercept is the point where the graph crosses the y-axis, and at this point, the value of \(x\) is zero. To find the y-intercept, substitute \(x = 0\) into the equation. For the given linear equation \(y = 3x + 3\), the process is straightforward:

• Substitute \(x = 0\) into the equation: \(y = 3(0) + 3\).
• Simplify to find \(y = 3\).

Therefore, the y-intercept is at the point \((0, 3)\). This tells us where the line meets the y-axis, which is a crucial part of graphing the function accurately. Knowing the y-intercept helps in drawing the line since it provides a starting point on the graph.

Creating a table of values is a practical method to plot several points of a linear function, ensuring precision in graphing. This technique involves selecting a range of \(x\) values and using the equation to find corresponding \(y\) values. For example, using \(y = 3x + 3\), you calculate:

• When \(x = -2\), then \(y = 3(-2) + 3 = -3\).
• When \(x = -1\), then \(y = 3(-1) + 3 = 0\).
• When \(x = 0\), then \(y = 3(0) + 3 = 3\).
• When \(x = 1\), then \(y = 3(1) + 3 = 6\).
• When \(x = 2\), then \(y = 3(2) + 3 = 9\).

Once you have these points \((-2, -3), (-1, 0), (0, 3), (1, 6), (2, 9)\), you can plot them on a coordinate graph. Drawing a line through these plotted points yields the linear function's graph, demonstrating how easily a table of values can facilitate accurate graphing of linear equations.