91Ó°ÊÓ

Graphing linear equations involves plotting points on a coordinate plane and connecting them to form a straight line. The equation given here is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the equation \(y = -x + 4\), the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(4\). To graph the equation, you must first create a table of values:

• Select a few values for \(x\), such as \(-2, -1, 0, 1,\) and \(2\).
• Calculate the corresponding \(y\) values using the equation \(y = -x + 4\). For instance, if \(x = -2\), then \(y = -(-2) + 4 = 6\).

Plot these points on the coordinate plane:

• \((-2, 6)\)
• \((-1, 5)\)
• \((0, 4)\)
• \((1, 3)\)
• \((2, 2)\)

When you plot these, draw a straight line through them, ensuring it extends across the graph to show the linear equation's nature. Remember, a linear equation like this represents an infinite line, so extend it beyond the points you plotted.

The x-intercept is a crucial point in graphing as it's where the line crosses the x-axis. To find the x-intercept of the equation \(y = -x + 4\), set \(y\) to zero and solve for \(x\). This is because, along the x-axis, the value of \(y\) is always zero:1. Start with the equation: \(y = -x + 4\)2. Set \(y = 0\) to find the x-intercept: \(0 = -x + 4\)3. Rearrange to solve for \(x\): - Move 4 to the left side: \(-x = -4\) - Divide by -1 to get \(x = 4\)The x-intercept, therefore, is the point \((4, 0)\). This means the graph crosses the x-axis at this point. Visually, this can also help check if your graph is accurate, as it should intersect the x-axis at this calculated intercept.

The y-intercept is the point where the graph crosses the y-axis. This occurs when \(x\) is zero because all points on the y-axis have an \(x\)-value of zero. To find the y-intercept for the equation \(y = -x + 4\), you simply substitute \(x = 0\):1. Use the equation: \(y = -x + 4\)2. Substitute \(x = 0\) into the equation: - \(y = -(0) + 4\) - Thus, \(y = 4\)Therefore, the y-intercept is at the point \((0, 4)\). On the graph, this is where the line will cross the vertical y-axis. This intercept is particularly significant as it's also the starting point when graphing the line using the slope-intercept form \(y = mx + b\). Look for this point when graphing to ensure your line is accurate.