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Understanding graphs of linear relationships

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Understanding Graphs of Linear Relations

A linear relation graphs as a straight line that encodes its y-intercept (where it crosses the y-axis) and its slope (rise over run, how steeply it climbs). For y equals 2x plus 1, the y-intercept is (0,1) and the slope is 2. Learn to read both directly off any linear graph.

What a linear relation graph shows

A linear relation graphs as a straight line, and that line encodes two pieces of information at a glance: where it crosses the y-axis, and how steeply it rises or falls. Learning to read both from the picture is the foundation for every other linear-graphing skill.

Reading a linear relation graph The graph of y equals 2x plus 1. It crosses the y-axis at 0,1 (the y-intercept), and for every 1 unit moved right, the line rises 2 units (the slope). xy y-intercept (0, 1) rise = 2 run = 1 y = 2x + 1 (slope = 2)
The line y = 2x + 1 crosses the y-axis at (0, 1) and rises 2 units for every 1 unit moved right.

The y-intercept

The y-intercept is the point where the line crosses the y-axis — the value of y when x = 0. For y = 2x + 1, that point is (0, 1). On any graph, it is the easiest point to spot: just look at where the line touches the vertical axis.

The slope

The slope measures steepness as "rise over run": how far the line moves up (or down) for a given move to the right. For y = 2x + 1, moving 1 unit right (run = 1) moves the line up 2 units (rise = 2), so the slope is 2/1 = 2. A steeper line has a larger slope; a line that falls has a negative slope.

From graph to table, and back

The same relation can be shown as a graph, an equation, or a table of values — all three describe the same line. Once you can read the intercept and slope off a graph, you are ready to build the graph from an equation instead, which is where introduction to linear equations and solving two-step linear equations pick up.

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