The slope-intercept form of a line is y equals m x plus b. Here, m represents the slope and b represents the y-intercept. Given a slope of two over two, which simplifies to one, and a y-intercept of one, the equation becomes y equals x plus one.
The y-intercept is the point where the line crosses the y-axis. For the equation y equals x plus one, this occurs at the point zero comma one. This is our starting point for graphing the line.
The slope of one means that for every one unit moved to the right along the x-axis, the line rises one unit along the y-axis. This is described as 'rise one, run one'. Starting from the y-intercept at zero comma one, we can find another point by moving up one and right one to reach the point one comma two.
Using the slope of one, we can find additional points on the line. Starting from zero comma one, moving right one and up one gives us one comma two, and again gives two comma three. Moving left one and down one from zero comma one gives negative one comma zero, and again gives negative two comma negative one.
Now that we have plotted several points, we can draw a straight line through them. This line represents all solutions to the equation y equals x plus one. The line extends infinitely in both directions, which we indicate with arrows at each end. The line passes through zero comma one, has a positive slope, and rises from left to right.
To verify our graph is correct, we can check that several points satisfy the equation y equals x plus one. For zero comma one: one equals zero plus one, which is true. For two comma three: three equals two plus one, which is true. For negative one comma zero: zero equals negative one plus one, which is also true. The key properties of this line are: a positive slope of one, meaning it rises at a forty-five degree angle, and a y-intercept of one, meaning it crosses the y-axis at zero comma one.