Math
1314 College
Algebra LO-5 Chapters 2.1, 2.2
Skills to Master
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Identify the
parts of the rectangular (Cartesian) coordinate system, including the x-axis,
y-axis, origin, and quadrants
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Plot and read
points from the rectangular (Cartesian) coordinate system
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Graph a linear
equation by creating a table of values
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Graph a linear
equation by finding the x- and y-intercepts
·
Graph a linear
equation in slope-intercept form by using the values of m and b
·
Graph horizontal
and vertical lines in the Cartesian plane
·
Use the graph of
a linear equation to find the slope of the line
·
Use two points
found on a line to determine its slope using the slope formula
·
Know the slope
of vertical and horizontal lines
·
Determine the
slope of a line parallel or perpendicular to a second line
·
The
Rectangular Coordinate System
Suppose that the above grid represented a
neighborhood with your house at the center, the grid lines representing
streets, and houses at each intersection of two grid lines. How might you give directions to someone who
wanted to get from your house to any other house in the neighborhood?
The above grid is called the coordinate plane or the Cartesian plane. It is basically two numbers lines: one horizontal number line
called the x-axis, and one vertical
number line called the y-axis. The point at which these two axes intersect
is called the origin
Plotting points on the plane is equivalent to giving
directions. But the “directions” are in
compact form: (x, y). This is called an ordered pair because the order is
important. To plot a point on the
coordinate plane, start at the origin
(0, 0), count x spaces horizontally, then from the point at which you stop,
count y spaces vertically. REMEMBER: the
numbers in an ordered pair represent ONE POINT, NOT TWO POINTS! Each number in an ordered pair is called a coordinate. The first value is the x-coordinate, and the second is the y-coordinate.
Plot the following point on the grid above: A (4,
0), B (-1, 0), C (0, -4), D (1, -2), E (-2, 1)
Are D and E the same or different points?
In this chapter you will study equations that have
the form
In the above examples, what are the values of m and b in each equation?
Since these equations have two different variables, it is
impossible to solve them in the same way we solved before. That is, we cannot end up with a single
number answer. This kind of equation
allows us to determine the values of one of the variables (called the dependent
variable) over all possible values of the other variable (called the
independent variable). By convention, we
choose y to be the dependent variable
and x to be the independent
variable. Since x is independent, we can choose any real number values for x and use the equation to determine the
corresponding values for y. Examples:
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Another form of a linear equation (sometimes called
the standard form) is
Notice that the x-intercept is always of the form (x, 0) and the y-intercept is of the form
(0, y). Therefore, we can find the x-intercept by
substituting zero for y in a linear
equation, and we can find the y-intercept by substituting zero for x in a linear equation.
Example
Find the x- and y-intercepts for the equation
Find the x- and y-intercepts for the equation
The benefit of using this method is that once you know two
points of the line, you have enough information to graph the entire line. Simply plot the points and draw the line
between them.
Example
The Slope of a Line
The slope
of a line is the measure of the slant of the line. A better and more precise way to think of the
slope of a line is the rate at which the y values change with respect to the x
values.
Two find the slope of a line from the graph of that
line, count the rise (the vertical
distance from one point to any other point on the line); from there count the run (the horizontal distance from one
point to any other point on the line); the slope is the ratio rise over run. In the above example, the rise is +5 and the
run is +1, so the slope of the above line is
Example
Find the slopes of each of the three lines below.
How can the slope be found if the graph is not
given? The slope of a line can be found
when two points on the line are given by using the formula
Slope of line one:
Slope of line two:
Slope of line three:
What is the slope of the line passing through the points (4, -9)
and (-8, -9)?
For this last example, notice that the slope is zero. The line above is a horizontal line. The
slope of a horizontal (flat) line is always zero. What about the slope of a vertical line? The points (6, 0) and (6, -3) are points on a
vertical line. Use the slope formula to
find the slope of this line.
The
slope of a vertical line is always undefined. This is true because the run in such a line
is zero. Since division by zero is
undefined, there is no real number associated with the slope. Thus we say that the slope is undefined.
Parallel and Perpendicular Lines
There is an important relationship between the slopes of
parallel and perpendicular lines:
1. Parallel
lines have equal slopes
2. Perpendicular
lines have slopes that are negative reciprocals of each other
Example
Find the slopes of the lines passing through the
given points:
Line 1: (4,
3) and (-1, -1)
Line 2: (5, 6) and (-10, -6)
Line 3: (4, -5) and (8, -10)
Which two lines are parallel? Which lines are perpendicular?
When given another line in slope-intercept form, the
slope of a second line can be found as follows:
1. To
find the slope of another line that is parallel to the given line, simply copy
the value of the slope from the equation given
2. To
find the slope of a second line that is perpendicular to the given line, take
the reciprocal of the slope of the given line and change its sign
Examples
Find the slope of a line that is parallel to
Find the slope of a line that is perpendicular to
Try these:
Find the slope of a line that is parallel to
Find the slope of a line that is perpendicular to