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If you ve taken a course in algebra or geometry, you ve learned about the graphing system called Cartesian (pronounced car-TEE-zhun ) two-space, also known as Cartesian coordinates or the Cartesian plane Let s review the basics of this system, and then we ll learn how to calculate distances in it
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We can put together a Cartesian plane by positioning two identical real-number lines so they intersect at their zero points and are perpendicular to each other The point of intersection is called the origin Each number line forms an axis that can represent the values of a mathematical variable
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The variables Figure 1-1 shows a simple set of Cartesian coordinates One variable is portrayed along a horizontal line, and the other variable is portrayed along a vertical line The number-line scales are graduated in increments of the same size Figure 1-2 shows how several ordered pairs of the form (x,y) are plotted as points on the Cartesian plane Here, x represents the independent variable (the input ), and y represents the dependent variable (the output ) Technically, when we work in the Cartesian plane, the numbers in an ordered pair represent the coordinates of a point on the plane People sometimes say or write things as if the ordered pair actually is the point, but technically the ordered pair is the name of the point Interval notation In pre-calculus and calculus, we ll often want to express a continuous span of values that a variable can attain Such a span is called an interval An interval always has a certain minimum value and a certain maximum value These are the extremes of the interval Let s be sure that
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Positive dependentvariable axis
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6 Negative independentvariable axis 4 2
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Negative dependent 6 variable axis
Positive independentvariable axis
Figure 1-1
The Cartesian plane consists of two realnumber lines intersecting at a right angle, forming axes for the variables
y 6 Origin = (0, 0) 4 2 x (4, 3)
( 4, 5)
2 2
( 5, 3)
4 6
Ordered pairs are of the form (x, y) (1, 6)
Figure 1-2
Five ordered pairs (including the origin) plotted as points on the Cartesian plane The dashed lines are for axis location reference
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you re familiar with standard interval terminology and notation, so it won t confuse you later on Consider these four situations: 0<x<2 1 y < 0 4<z 8 p q p These expressions have the following meanings, in order: The value of x is larger than 0, but smaller than 2 The value of y is larger than or equal to 1, but smaller than 0 The value of z is larger than 4, but smaller than or equal to 8 The value of q is larger than or equal to p, but smaller than or equal to p
The first case is an example of an open interval, which we can write as x (0,2) which translates to x is an element of the open interval (0,2) Don t mistake this open interval for an ordered pair! The notations look the same, but the meanings are completely different The second and third cases are examples of half-open intervals We denote this type of interval with a square bracket on the side of the included value and a rounded parenthesis on the side of the non-included value We can write y [ 1,0) which means y is an element of the half-open interval [ 1,0), and z (4,8] which means z is an element of the half-open interval (4,8] The fourth case is an example of a closed interval We use square brackets on both sides to show that both extremes are included We can write this as q [ p,p] which translates to q is an element of the closed interval [ p,p]
Relations and functions Do you remember the definitions of the terms relation and function from your algebra courses (If you read Algebra Know-It-All, you should!) These terms are used often in pre-calculus, so it s important that you be familiar with them A relation is an operation that transforms, or maps, values of a variable into values of another variable A function is a relation in which there is never more than one value of the dependent variable for any value of the independent variable In other words, there can t be more than one output for any input (If a particular input