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- Tidymymusic 3 0 1 2 X 42
- Tidymymusic 3 0 1 2 X 44
- Tidymymusic 3 0 1 2 X 4.5
- Tidymymusic 3 0 1 2 X 4 1 2
The Coordinate Plane
Learning Objective(s)
·Plot ordered pairs on a coordinate plane.
·Given an ordered pair, determine its quadrant.
The coordinate plane was developed centuries ago and refined by the French mathematician René Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. The coordinate plane can be used to plot points and graph lines. This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.
Getting to Know the Coordinate Plane
You have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object? (This is often done with road maps, too.)
This 'map' uses a horizontal and vertical grid to convey information about an object's location. Notice that the letters A-F are listed along the top, and the numbers 1-6 are listed along the left edge. The general location of any item on this map can be found by using the letter and number of its grid square. For example, you can find the item that exists at square '4F' by moving your finger along the horizontal to letter F and then straight down so you are in line with the 4. You'll find a blue disc is at this location on the map.
The coordinate plane has similar elements to the grid shown above. It consists of a horizontal axis and a vertical axis, number lines that intersect at right angles. (They are perpendicular to each other.)
The horizontal axis in the coordinate plane is called the x-axis. The vertical axis is called the y-axis. The point at which the two axes intersect is called the origin. The origin is at 0 on the x-axis and 0 on the y-axis.
The intersecting x- and y-axes divide the coordinate plane into four sections. These four sections are called quadrants. Quadrants are named using the Roman numerals I, II, III, and IV beginning with the top right quadrant and moving counter clockwise.
Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a point by relating the point's location along the x-axis (the first value of the ordered pair) and along the y-axis (the second value of the ordered pair).
In an ordered pair, such as (x, y), the first value is called the x-coordinate and the second value is the y-coordinate. Note that the x-coordinate is listed before the y-coordinate. Since the origin has an x-coordinate of 0 and a y-coordinate of 0, its ordered pair is written (0, 0).
Consider the point below.
To identify the location of this point, start at the origin (0, 0) and move right along the x-axis until you are under the point. Look at the label on the x-axis. The 4 indicates that, from the origin, you have traveled four units to the right along the x-axis. This is the x-coordinate, the first number in the ordered pair.
Tidymymusic 3 0 1 2 X 42
From 4 on the x-axis move up to the point and notice the number with which it aligns on the y-axis. The 3 indicates that, after leaving the x-axis, you traveled 3 units up in the vertical direction, the direction of the y-axis. This number is the y-coordinate, the second number in the ordered pair. With an x-coordinate of 4 and a y-coordinate of 3, you have the ordered pair (4, 3).
Let's look at another example.
Example | |
Problem | Describe the point shown as an ordered pair. |
(5, y) | Begin at the origin and move along the x-axis. This is the x-coordinate and is written first in the ordered pair. |
(5, 2) | Move from 5 up to the ordered pair and read the number on the y-axis. This is the y-coordinate and is written second in the ordered pair. |
Answer | The point shown as an ordered pair is (5, 2). |
Now that you know how to use the x- and y-axes, you can plot an ordered pair as well. Just remember, both processes start at the origin—the beginning! The example that follows shows how to graph the ordered pair (1, 3).
Example | |
Problem | Plot the point (1, 3). |
The x-coordinate is 1 because it comes first in the ordered pair. Start at the origin and move a distance of 1 unit in a positive direction (to the right) from the origin along the x-axis. | The y-coordinate is 3 because it comes second in the ordered pair. From here move directly 3 units in a positive direction (up). If you look over to the y-axis, you should be lined up with 3 on that axis. |
Answer | Draw a point at this location and label the point (1, 3). |
In the previous example, both the x- and y-coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.
Example | |
Problem | Plot the point (−4, −2). |
The x-coordinate is −4 because it comes first in the ordered pair. Start at the origin and move 4 units in a negative direction (left) along the x-axis. | The y-coordinate is −2 because it comes second in the ordered pair. Now move 2 units in a negative direction (down). If you look over to the y-axis, you should be lined up with −2 on that axis. |
Answer | Draw a point at this location and label the point (−4, −2). |
The steps for plotting a point are summarized below.
Steps for Plotting an Ordered Pair (x, y) in the Coordinate Plane oDetermine the x-coordinate. Beginning at the origin, move horizontally, the direction of the x-axis, the distance given by the x-coordinate. If the x-coordinate is positive, move to the right; if the x-coordinate is negative, move to the left. oDetermine the y-coordinate. Beginning at the x-coordinate, move vertically, the direction of the y-axis, the distance given by the y-coordinate. If the y-coordinate is positive, move up; if the y-coordinate is negative, move down. oDraw a point at the ending location. Label the point with the ordered pair. |
Which point represents the ordered pair (−2, −3)? |
The Four Quadrants
Ordered pairs within any particular quadrant share certain characteristics. Look at each quadrant in the graph below. What do you notice about the signs of the x- and y-coordinates of the points within each quadrant?
Within each quadrant, the signs of the x-coordinates and y-coordinates of each ordered pair are the same. They also follow a pattern, which is outlined in the table below.
Quadrant | General Form of Point in this Quadrant | Example | Description |
I | (+, +) | (5, 4) | Starting from the origin, go along the x-axis in a positive direction (right) and along the y-axis in a positive direction (up). |
II | (−, +) | (−5, 4) | Starting from the origin, go along the x-axis in a negative direction (left) and along the y-axis in a positive direction (up). |
III | (−, −) | (−5, −4) | Starting from the origin, go along the x-axis in a negative direction (left) and along the y-axis in a negative direction (down). |
IV | (+, −) | (5, −4) | Starting from the origin, go along the x-axis in a positive direction (right) and along the y-axis in a negative direction (down). |
Once you know about the quadrants in the coordinate plane, you can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here's another way to think about it.
The example below details how to determine the quadrant location of a point just by thinking about the signs of its coordinates. Thinking about the quadrant location before plotting a point can help you prevent a mistake. It is also useful knowledge for checking that you have plotted a point correctly.
Example | |
Problem | In which quadrant is the point (−7, 10) located? |
(−7, 10) | Look at the signs of the x- and y-coordinates. For this ordered pair, the signs are (−, +). |
Points with the pattern | Using the table or grid above, locate the pattern (−, +). |
Answer | The point (−7, 10) is in Quadrant II. |
Example | |
Problem | In which quadrant is the point (−10, −5) located? |
(−10, −5) | Look at the signs of the x- and y-coordinates. For this ordered pair, the signs are (−, −). |
Points with the pattern | Using the table or grid above, locate the pattern (−, −). |
Answer | The point (−10, −5) is in Quadrant III. |
What happens if an ordered pair has an x- or y-coordinate of zero? The example below shows the graph of the ordered pair (0, 4).
A point located on one of the axes is not considered to be in a quadrant. It is simply on one of the axes. Whenever the x-coordinate is 0, the point is located on the y-axis. Similarly, any point that has a y-coordinate of 0 will be located on the x-axis.
Which of the descriptions below best describes the location of the point (8, 0)? A) Quadrant I B) It is on the x-axis C) It is on the y-axis D) The coordinate plane |
Summary
The coordinate plane is a system for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (x-) axis and a vertical (y-) axis. The intersection of these lines creates the origin, which is the point (0, 0). The coordinate plane is split into four quadrants. Together, these features of the coordinate system allow for the graphical representation and communication about points, lines, and other algebraic concepts.
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Composition of Functions:
Composing Functions at Points (page 2 of 6)
Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition
Tidymymusic 3 0 1 2 X 44
Suppose you are given the two functions f (x) = 2x + 3 and g(x) = –x2 + 5. Composition means that you can plug g(x) into f (x). This is written as '( fog)(x)', which is pronounced as 'f-compose-g of x'. And '( fog)(x)' means ' f (g(x))'. That is, you plug something in for x, then you plug that value into g, simplify, and then plug the result into f. The process here is just like what we saw on the previous page, except that now we will be using formulas to find values, rather than just reading the values from lists of points.
- Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (gof )(1).
When I work with function composition, I usually convert '( fog)(x)' to the more intuitive ' f (g(x))' form. This is not required, but I certainly find it helpful. In this case, I get:
(gof )(1) = g( f(1))
This means that, working from right to left (or from the inside out), I am plugging x = 1 into f(x), evaluating f(x), and then plugging the result into g(x). I can do the calculations bit by bit, like this: Sincef(1) = 2(1) + 3 = 2 + 3 = 5, and since g(5) = –(5)2 + 5 = –25 + 5 = –20, then (gof )(1) = g( f(1)) = g(5) = –20. Doing the calculations all together (which will be useful later on when we're doing things symbolically), it looks like this:
(gof )(1) = g( f (1))
= g(2( ) + 3) .. setting up to insert the original input
= g(2(1) + 3)
= g(2 + 3)
= g(5)
= –( )2 + 5 .. setting up to insert the new input
= –(5)2 + 5
= –25 + 5
= –20
Note how I wrote each function's rule clearly, leaving open parentheses for where the input (x or whatever) would go. This is a useful technique. Whichever method you use (bit-by-bit or all-in-one), the answer is:
(gof )(1) = g( f (1)) = –20
I just computed (gof )(1); the composition can also work in the other order:
Tidymymusic 3 0 1 2 X 4.5
- Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( fog)(1).
First, I'll convert this to the more intuitive form, and then I'll simplify:
( fog)(1) = f (g(1))
Working bit-by-bit, since g(1) = –(1)2 + 5 = –1 + 5 = 4, and since f(4) = 2(4) + 3 = 8 + 3 = 11, then ( fog)(1) = f (g(1)) =f(4) = 11. On the other hand, working all-in-one (right to left, or from the inside out), I get this:
( fog)(1) = f (g(1))
= f (–( )2 + 5) .. setting up to insert the original input
= f (–(1)2 + 5)
= f (–1 + 5)
= f (4)
= 2( ) + 3 .. setting up to insert the new input
= 2(4) + 3
= 8 + 3
= 11
Either way, the answer is: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
( fo g)(1) = f (g(1)) = 11
A verbal note: 'fog' is not pronounced as 'fogg' and 'gof ' is not pronounced as 'goff'. They are pronounced as 'f-compose-g' and 'g-compose-f', respectively. Don't make yourself sound ignorant by pronouncing these wrongly!
As you have seen above, you can plug one function into another. You can also plug a function into itself:
Tidymymusic 3 0 1 2 X 4 1 2
- Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( fof )(1).
( fof )(1) = f ( f (1))
= f (2( ) + 3) .. setting up to insert the original input
= f (2(1) + 3)
= f (2 + 3)
= f (5)
= 2( ) + 3 .. setting up to insert the new input
= 2(5) + 3
= 10 + 3
= 13
- Givenf(x) = 2x + 3 and g(x) = –x2 + 5, find (gog)(1).
Dvd cloner 6 40 – copycloneburn dvd movies on mac. (gog)(1) = g(g(1))
= g(–( )2 + 5) .. setting up to insert the original input
= g(–(1)2 + 5)
= g(–1 + 5)
= g(4)
= –( )2 + 5 .. setting up to insert the new input
= –(4)2 + 5
= –16 + 5
= –11
In each of these cases, I wrote out the steps carefully, using parentheses to indicate where my input was going with respect to the formula. If it helps you to do the steps separately, then calculate g(1) outside of the other g(x) as a separate step. That is, do the calculations bit-by-bit, first finding g(1) = 4, and then plugging 4 into g(x) to get g(4) = –11.
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Cite this article as: | Stapel, Elizabeth. 'Composing Functions at Points.' Purplemath. Available from |
