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Linear Functions and Slope. What is slope? What is slope? The steepness of the graph, the rate at which the y values are changing in relation to the changes in x . How do we calculate it?. A line has one slope. Between any 2 pts. on the line, the slope MUST be the same.

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What is slope? What is slope?The steepness of the graph, the rate at which the y values are changing in relation to the changes in x.How do we calculate it?

a line has one slope
A line has one slope
  • Between any 2 pts. on the line, the slope MUST be the same.
  • Use this to develop the point-slope form of the equation of the line.
  • Now, you can develop the equation of any line if you know either a) 2 points on the line or

b) one point and the slope.

find the equation of the line that goes through 2 5 and 3 4
Find the equation of the line that goes through (2,5) and (-3,4)

1st: Find slope of the line

m=

2nd: Use either point to find the equation of the line & solve for y.

slope intercept form of the equation of the line
Slope-Intercept Form of the Equation of the Line
  • It is often useful to express the line in slope-intercept form, meaning that the equation quickly reveals the slope of the line and where it intercepts the y-axis.
  • It is REALLY using the point-slope form, except that the point is the intercept, (0,b).
  • y - b=m(x - 0) becomes y = mx + b
horizontal vertical lines
Horizontal & Vertical lines
  • What is the slope of a vertical line?
    • It is INFINITELY steep, it only rises. It is SO steep, we can’t define it, therefore undefined slope.
    • Look at some points on the line x = -4, i.e. (-4,8), (-4,9),(-4,13),(-4,0). We don’t care what value y has, all that matters is that x = - 4. Therefore, that is the equation of the line.
  • What is the slope of a horizontal line?
    • There is no rise, it only runs, the change in y is zero, so the slope = 0.
    • Look at some points on the line y = 5, i.e., (2,5), (-2,5), (17,5), etc. We don’t care what value x has, all that matters is that y = 5. Therefore, that is the equation of the line.
parallel lines
Parallel lines
  • Slopes are equal.
  • If you are told a line is parallel to a given line, you automatically know the slope of your new line (same as the given).
  • Find the equation of the line parallel to y=2x-7 passing through the point (3,-5).
    • slope = 2, passes through (3,-5)

y -(-5) = 2(x – 3)

y + 5 = 2x – 6

y = 2x - 11

perpendicular lines
Perpendicular lines
  • Lines meet to form a right angle.
  • If one line has a very steep negative slope, in order to form a right angle, it must intersect another line with a gradual positive slope.
  • The 2 lines graphed here illustrate that relationship.
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What about the intersection of a horizontal line and a vertical line? They ALWAYS intersect at a right angle. Since horizontal & vertical lines are neither positive or negative, we simply state that they are, indeed, ALWAYS perpendicular.
  • What about all other lines? In order to be perpendicular, their slopes must be the negative reciprocal of each other. You could also say that the product of the slopes must be -1.
find the equation of the line perpendicular to y 2 x 7 through the point 2 7
Find the equation of the line perpendicular to y = 2x – 7 through the point (2,7) .
  • y = ½ x + 7
  • y = - 2 x + 1/7
  • y = - ½ x + 4/3
  • y = - ½ x + 8
average rate of change
Average Rate of Change
  • Slope thus far has referred to the change of y as related to the change in x for a LINE.
  • Can we have slope of a nonlinear function?
  • We CAN talk about the slope between any 2 points on the curve – this is the average rate of change between those 2 points.