From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes Linear Functions A linear function in two variables is any equation of that may be written in the form y = mx + b where m and b are real number coefficients and x and y represent any real numbers that make up a solution. Furthermore, we observe that • The point (0, b) will always be the y-intercept. • The slope of the line will always equal m. • The slope is defined as m = (y2– y1)/(x2– x1)for any two points on the line. We call y = mx + b the Slope-Intercept Form of the linear equation. Slope The slope of a line is defined descriptively as the ratio of how far up you move divided by how far to the right you move, as you move from one point to any other point on the line. Example:The graph of y = 2/3x – 1 has a slope of 2/3 and a y-intercept of (0,-1) as shown below. Negative Slope If we must move down instead of up, a negative sign is part of the slope. Also, if we must move left instead of right, a negative sign is part of the slope. So if slope is negative, we can interpret this in one of two ways: • m = -(A/B) = (-A)/B which would indicate that you move down A units and then right B units. • m = -(A/B) = A/(-B) which would indicate that you move up A units and then left B units. Note that if we must move both down and left when moving from point to point, two negative signs are incorporated into the slope and the result is a positive ratio.
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From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes Example: The graph of y = -2x +1 indicates a slope of –2. We may write m = -2 as m = (-2)/1. So we move down 2 and right 1 when moving from point to point as shown below. Alternatively, we could write this slope as m = 2/(-1) and move up 2 and left 1 when moving from point to point. Example:Calculate the slope of the line passing through (4,-1) and (2, -2). Then use this slope to help graph this line. The slope is given by m = (y2– y1)/(x2– x1). So we substitute in our values to get m = -2 – (-1) = -2 + 1 = -1= 1 2 – 4 -2 -2 2 The graph is given by