Lesson 20 MA 152, Section 2.2, 2.3 (part 1)
Slope of a line: If a non-vertical line contains points , the slope of the line is the ratio described by .
*Note: Always be consistent in the order of the coordinates.
This is the first of 3 ways you can find the slope of a line. The 3 ways are:
- Using the slope formula (above)
- Counting rise over run (when shown a graph)
- Solving the equation for y (discussed in section 2.3)
Ex 1: Find the slope of a line containing each pair of points.
Ex 2: Find the slope of each line given its equation.
If a line is horizontal, the numerator in the slope formula will be 0 (the y coordinates of all points of a horizontal line are the same). The slope of a horizontal line is 0.
If a line is vertical, the denominator in the slope formula will be 0 (the x coordinates of all points of a vertical line are the same). A number with a zero denominator is not defined or undefined. The slope of a vertical line is not defined.
There are 4 types of slopes.
Positive Negative Zero Undefined
Parallel Lines: Two lines that are parallel will have the same slope or two lines with the same slope will be parallel.
Perpendicular lines: Two lines that are perpendicular will have slopes with a product of -1 (opposite reciprocals or negative reciprocals). Two lines whose slopes of negative reciprocals will be perpendicular.
Ex 3: Determine is the lines with given slopes or given pairs of points are parallel, perpendicular, or neither (simply intersect).
Ex 4: Determine if the following points are vertices of a right triangle.
The average rate of change for a problem where data is predicated to be a straight line pattern is the slope.
Ex 5: A small business predicts sales according to a straight line method. If sales were $50,000 in the first year and $110,000 in the third year, find the rate of growth in dollars per year.
Section 2.3:
There are 3 forms for the equation of a line. We have already mentioned a linear equation has the form This is called general form.
General Form
In general form A, B, and C are integers and A is positive.
The next we will cover is the point-slope form, which is derived from the slope formula.
Point-Slope Form
Ex 5: a) Write an equation in point-slope form for a line with a slope of and through the point (2, 12).
b) Find the slope and an indicated point for a line with equation
.
Ex 6: Find the equation of each line described in general form.
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