The Science That Draws Necessary Conclusions

A linear function is a function whose graph is a line. They are written in three forms: slope-intercept form, point-slope form, and general form.

Slope-intercept is a common way to write a linear function. Its form is . The graph of is shown in the image above. The variables m and b modify this graph.

m is the slope of the line (how steep the line is). In , m is equal to 1. As a result, one point can be reached from another point by moving up 1 unit and to the right 1 unit. The function has a slope of 2 so it will move up 2 units and to the right 1 unit. The function has a slope of so it will move up 1 unit and to the right 2 units. When m is negative, the line will be moving down rather than up.

b is the y-intercept of the line (where the line intersects the y-axis). The y-intercept of is 0. As a result, the line crosses the y-axis at the origin. The line crosses the y-axis at -3.

Point-slope form is . This can be written as a function as (y become f(x)). and are the coordinates of a point on the graph. This can be useful if the slope and a point on the graph is known. For example, a line has a slope of 3 and passes through the point (3, 5). with the point and slope plugged in becomes . After distributing the 3 and adding the -9 to the 5 gives the graphable slope-point form .

General form is . It is less commonly used and at the moment not that important. It can be converted to point-slope form by changing it to . The slope is and the y-intercept is .

The graph above can be written as (slope-point), (point-slope), or as (general form).

A relation is composed of ordered pairs of numbers. The set of the first numbers is the domain and the set of the second values is the range of the relation. The first number is called the abscissa and the second number is called the ordinate.

The ordered pairs can be graphed using the Cartesian coordinate system. There are two axes in the coordinate system, the x and y axis. The x-axis is horizontal and the y-axis is vertical. The point where the lines cross is called the origin. That point is (0, 0). The further a point is toward the right, the larger the abscissa is. Moving toward the left from the origin goes negative. The further the point is upward, the larger the ordinate of the point is. The abscissa can be called the x value and the ordinate can be called the y value.

A function is a special relation where each abscissa has no more than one ordinate. This means that any input will always yield the same output. Note that different inputs may share the same output. The input is the independent variable x and the output is the dependent variable y.

The vertical line test tests whether a relation is a function. If a vertical line can be placed on a graph and it crosses through two or more points, then the line is not a function. The graph of a circle is not a function because a vertical line can be drawn that passes through more than one point. The picture of the graph on the right is a function because no matter where a vertical line is placed, it will only pass through one point. Each x value has only one y value.

A functions is written as . f(x) is the same as the y coordinate. So if , then and (3, 9) is a point of the graph of f(x). There are numerous functions to learn, but luckily there are patterns.

Sorry for the lack of lessons over the past week and a half. I have been on vacation, but now it is time to get back to work. I will post the test answers. Then I will either do some posts on algebra, functions and graphing, or continue with more on trigonometry.

I have made an assessment for what was taught so far. You can see it in the sidebar on the right or click on this. Unfortunately, I realized I forgot to mention the relevance of which quadrant the angle lies in. I will edit that into the special angles post. The answers will be posted in a few days. I am not yet sure where I will put them.

Before I create a test for the posts up to now, it is important for me to discuss coterminal and reference angles.

There are two parts of an angle, the initial side and terminal side. As the names imply, the initial side is the side where an angle begins and the terminal side is where the angle ends. The vertex is the point where the two sides meet. One can graph an angle on the coordinate plane. Standard position is when the vertex is at the origin (0, 0) and the initial side lies on the x-axis going in the positive direction. If the terminal side is moved counterclockwise, it is positive; if it is moved clockwise, it is a negative angle. Angles can be of any size, even greater than . An arrow normally notes the number of rotations, though it is not found in the picture above.

Coterminal angles are angles whose terminal sides lay upon one another but are not the same angle. For example, , , and are coterminal angles because their terminal sides lay on each other. The reference angle of an angle is the distance of its terminal side from the x-axis. Angles from to have a reference angle the same as the angle. So the reference angle of is .

The following only applies to angles between and . If the terminal side lies in the second quadrant, the reference angle is . If the terminal side lies in the third quadrant, the reference angle is . If the terminal side lies in the fourth quadrant, the reference angle is . The quadrantal angles, , , , and , have no reference angle.

To find the reference angle of other angles, it is easiest to find a coterminal angle between and . Simply add or subtract 360 until you find it.

For those who have graphing calculator, I made a program that you can use with it.

:Input ":", A
:
:While A>360
:A-360A
:End
:
:While A0
:A+360A
:End
:
isp "COTERMINAL:",A
:
:If (A=90 or A=180 or A= 270 or A=360)
:Then
isp "QUADRANT ANGLE"
:Stop
:End
:
:If (A>270 and A180 and A90 and A<180)
:180-AA
:
isp "REF ANGLE:",A


It may not be the best, but it works. You may want to add a “DelVar A” at the end. The next post should be announcing a Trig Test Part 1. The actual test will be on a page listed in the sidebar.