91Ó°ÊÓ

Understanding the point-slope formula is key to graphing a line given a point and the slope. The point-slope formula is \( y - y_1 = m(x - x_1) \). Here,

• \( (x_1, y_1) \) is the given point on the line
• \(m\) is the slope of the line

With this formula, you can create an equation for the line quickly by substituting the given point and slope values directly into it. For instance, if you have a point \( (2, 3) \) and a slope \(m = -1\), just plug these values into the formula to get \( y - 3 = -1(x - 2) \). This is your point-slope form equation. You can then simplify and convert it into other useful forms.

Another crucial concept is the slope-intercept form. It is usually more straightforward for graphing and visualizing. The slope-intercept form is given by \( y = mx + b \), where:

• \(m\) is the slope
• \(b\) is the y-intercept

To convert from the point-slope form to the slope-intercept form, you need to simplify the equation. For example, given \( y - 3 = -1(x - 2) \), distribute and simplify to get: \( y - 3 = -x + 2 \). Adding 3 to both sides gives \( y = -x + 5 \). This equation in slope-intercept form clearly shows the slope \(m = -1\) and the y-intercept \(b = 5\). This makes it easier to immediately start plotting on a graph.

Plotting the points correctly is essential to drawing an accurate graph of a line. Start by plotting the given point, which in this example is \( (2, 3) \). Locate this point on the graph and mark it. Next, use the slope to find another point.

• The slope \(m = -1\) implies a change of \( -1 \) in y for every \( +1 \) in x.

From the point \( (2, 3) \), move 1 unit to the right along the x-axis (to \(x = 3\)), and 1 unit down along the y-axis (to \(y= 2\)) to find the next point \( (3, 2) \). Plot this point, too. Now, draw a straight line through these points. This line represents the equation of the line given the point and slope. Remember, the slope helps dictate the steepness and direction of the line, making it quite simple to plot once you get the hang of it.