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Slope-Intercept Form
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Sometimes the slope of a straight line is informally called rise over run This notion works as long as the independent variable is on the horizontal axis, the dependent variable is on the vertical axis, and we move to the right
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Try two points! Suppose we see a line in the Cartesian plane, and are able to locate two points on it and determine their exact coordinates:
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(x1, y1) = ( 2, 4) and (x2, y2) = (3, 5) The slope is ( y2 y1), which we call y, divided by (x2 x1), which we call x : m = y / x = (y2 y1)/(x2 x1) = (5 4)/[3 ( 2)] = 1/(3 + 2) = 1/5 This situation is illustrated in Fig 15-1
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y 6 ( 2,4) (3,5)
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2 x 6 4 2 2 4 6 m= y/ x = (5 4) / [3 ( 2)] = 1/5 2 4 6
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Figure 15-1 The slope of a line can be calculated from
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the coordinates of two points on that line
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238 Graphs of Linear Relations
Switching the order We can switch the points (x1, y1) and (x2, y2) and still get the same slope when we calculate it as above That s because both the numerator and the denominator end up being additive inverses (exact negatives) of what they were before Let s take
(x1, y1) = (3, 5) and (x2, y2) = ( 2, 4) The slope is again equal to y / x Calculating, we get m = y / x = (y2 y1)/(x2 x1) = (4 5)/( 2 3) = 1/( 5) = 1/5 When we know the coordinates of two points on a line, we can figure the slope going from the first point to the second, or going from the second point to the first; it doesn t matter But we must be careful not to confuse the coordinates We can reverse the external sequence in which we work with the points, but we can t reverse the internal sequence of either of the ordered pairs defining those points!
What is the intercept When we talk about the SI form of a straight line in the Cartesian plane, the term intercept refers to the value of a variable at the point where the line crosses the axis for that variable If y is the dependent variable, then we often talk about the y-intercept That s what is usually meant when we work with the SI form of an equation when graphing it in the xy-plane Two examples are shown in Fig 15-2 An intercept can be thought of as an ordered pair where one of the values (the one on the axis not being intercepted) is 0 We can plug 0 into a linear equation for one of the variables, and solve for the remaining variable to get its intercept This method can be more convenient than rearranging everything into SI form or drawing a graph, but all by itself it doesn t give us any visual reinforcement of the situation Putting it together In Chap 12, you learned the standard form for a first-degree equation in one variable If the variable is x, the standard form is
ax + b = 0
Slope-Intercept Form y 6 4 2 y-intercept is 3 x 6 4 2 2 4 6 2 4 6 Slope is positive
y-intercept is 2
Slope is negative
Figure 15-2 Two examples of y-intercept points for
straight lines The line that ramps upward as we move to the right has positive slope; the line that ramps downward as we move to the right has negative slope
where a and b are constants If you substitute y for 0 and then transpose the left and right sides, you get an equation for a linear function where y is the dependent variable and x is the independent variable: y = ax + b As things work out, the constant a is the slope of the graph, and the constant b is the y-intercept Because the slope is usually symbolized by m instead of a, you can write y = mx + b This is the classical expression of the SI form for a linear function