Chapter 2, Section 3 Slope

Reviewing types of slope Remember, there are four types of slope: Positive slope, negative slope, no slope and undefined slope. Positive Negative No slope Undefined slope

Definition of Slope Slope is defined as the ratio of the change in y to the change in x. We can represent this several ways: Where the capital Greek letter Delta (Δ) means “difference” or “change in”. The notation used in calculus for the first derivative -- the slope.

And the one WE will use in Alg 2: where m is our old friend from Algebra 1, when we put linear equations in slope intercept form, y = mx + b -- and m was the slope. This equation assumes we have two points, point one: P1(x1,y1), and point two: P2(x2,y2).

Finding the slope of a line from two points Given a graph with two points, we can calculate the slope of a line between them using the formula. It doesn’t matter which point we call point one or point two, but we MUST make sure we are consistent with which is which. Using point A as point 1 and point B as point 2: Using point B as point 1 and point A as point 2:

Making sense of “no” and “undefined” slopes From the formula, one can readily see that in a horizontal line there is no rise and lots of run, so the rise over run is zero over one, or zero. That’s NO slope. From the formula, one can also see that a vertical line has all rise and no run, and since that’s lots of rise over NO run, that’s division by zero, which is UNDEFINED.

Example 1 Find the slope of the line that passes through (1,3) and (-2,-3). Then graph the line. First we’ll calculate slope using the formula: Then we’ll graph the line from the given points.

Example 2 Graph the line through (1,-3) with a slope of -¾. First, graph the point (1,-3). Then use the two equivalent fractions for slope, or rise over run:

Related slopes Remember that two lines that are parallel have the same slope. Remember also that the product of two slopes that are perpendicular is negative one; i.e. one is the negative reciprocal of the other. What is the slope of a line parallel and perpendicular to the line y = (3/5)x - 2? Parallel: 3/5 Perpendicular: -(5/3) What is the slope of a line parallel and perpendicular to the line y = -(2/7)x + 5? Parallel: -(2/7) Perpendicular: 7/2

Last example Graph the line through (2,1) that is perpendicular to the line with equation 2x - 3y = 3. Since we have a point, we only need the slope to graph a line, so we’ll put the equation into slope-intercept form: 2x - 3y = 3 -3y = -2x + 3 y = (2/3)x - 1 So the slope of the line is (2/3). The slope of the line perpendicular to it is… (answer?) Answer:

Your assignment… Chapter 2 section 3, 15-28, 31-36, 43-48, 58-66, 68-75. The best revenge is to be unlike him who performed the injury. --Marcus Aurelius