Lesson 14MA 152, Sections 2.2 & 2.3
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.
PositiveNegativeZeroUndefined
Parallel Lines: Two lines that are parallel will have the same slope or two lines will 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 standard form.
Standard Form
In standard 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 standard form.
Ex 7:Find the equation of the line through (4, 6) and with a slope of and simplify.
Ex 8: Find the equation of the line through points and simplify.
Suppose the point given for a line is the y-intercept. It can be called (0, b).
This form for an equation of a line is called slope-intercept form, where m is the slope of the line and b represent the point (0,b), the y-intercept of the line.
Slope-intercept Form
Ex 9:Find an equation of a line with slope .
*Note: The third way to find slope is to put the equation in slope-intercept form.
Ex 10:Find the slope and y-intercept for a line .
Slope-intercept form can also be used to find the equation of a line if the point known is not the y-intercept (by solving for b). Examine this example.
Find the equation in standard form of a line with .
Ex 11:Find the equation of the line described in standard form by using the process above.
Ex 12:Determine if the following lines are parallel, perpendicular, or neither.
Ex 13:Find the equation in slope-intercept form for each line described.
Some textbooks, including the one we are using, also refer to standard form as general form. Examine the following.
This shows another way that the slope and y-intercept can be found from standard or general form.
Ex 14:Find the slope and y-intercept of the following, then write the equation in slope- intercept form.
Application Problems:
Ex 15:Steven has an antique watch that has appreciated in value (straight line pattern) from the time he purchased it. He bought the watch for $900. After 6 years, it was worth $1150. Write an equation of its value after t years.
Ex 16:Mary's car has been depreciating in value in a straight line pattern. Two years after she purchased the car new, its blue book value was $10,800. Five years after purchase, its value was $9000. Find an equation of its value for t years after it was new.
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