ࡱ> 463i @ ,bjbj)) "TKzKz$zzzzzzz8lj,)'V"xxx'''&&&&&&&$(R+b&z |'  &zzxx3&OOO zxzx&O &OOV#@zz$x @zX$$ &&0)'0$x}+}+$D,zzzz$8}+z$'"IOau'''&&DB?BAlgebra II Chapter 2 Notes: Linear Relations and Functions Section 2.1: Relations and Functions Cartesian coordinate plane: is composed of the x-axis (horizontal) and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into four quadrants. The quadrants are denoted by roman numerals going counter clockwise, so there is no confusion with points. History: The Cartesian coordinate system was developed by the French mathematician Rene Descartes during an illness (early 1600s). As he lay in bed sick, he saw a fly buzzing around on the ceiling, which was made of square tiles. As he watched he realized that he could describe the position of the fly by the ceiling tile he was on. After this experience he developed the coordinate plane to make it easier to describe the position of objects. He put 2 number lines together to create his system. Real-World Example: plotting points on the Cartesian Coordinate System is like playing notes on the piano or an instrument. The musician must understand the notes, rhythm and any sharps or flats in order to correctly play the music. The different clefs tell the musician which keys to play on the piano and in which position. Ordered pairs: (x, y) pairing of xs and ys. X comes first because it comes before y in the alphabet. Y comes last because it comes after x in the alphabet. Relation: a set of ordered pairs. Domain: of a relation is the set of all first coordinates (x-coordinates) from the ordered pairs. Range: is the set of all second coordinates (y-coordinates) from the ordered pairs. The graph of a relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation. Function: is a special type of relation in which each element of the domain is paired with exactly one element of the range. One-to-one function: where each element of the range is paired with exactly one element of the domain. X values CANNOT repeat! Only the y values can repeat. 2 Ways to Determine if a Relation is a Function 1. Mapping: shows how each member of the domain is paired with each member of the range. A map that shows how each x value is paired with the y values. 2. Vertical Line test: determines whether a relation is a function. If the vertical line does not intersect a graph in more than one point, the graph represents a function. If the vertical line intersects the graph at two or more points, the graph does NOT represent a function. Examples of graphs that are NOT functions: circles, hyperbolas, ellipses, semi-circles. Therefore, the vertical line can only hit the graph ONCE to be a function! Functional notation: f(x) = 2x + 1, the symbol f(x) is read f of x, the f is the name of the function, it not a variable that is multiplied by x. Section 2.2: Linear Equations (Omit Algebra II) Linear equation: contains no other operations than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in the denominator. It does not contain variables with exponents other than 1. Examples: x, 3, x + 5, x 5, 5x, .5x. Real-World Examples: Most data that we see on the television or newspaper has been formatted to a linear equation since this is the most recognizable format. You can determine the temperature if you listen to crickets chirping, a linear model has been developed. Cell phone plans can also be viewed linearly. Linear function: a function whose ordered pairs satisfy a linear equation, it can be written in the form f(x) = mx + b, where m and b are real numbers. Standard form: and linear equation can be written in the form: Ax + By = C, where A,B,C, are integers whose greatest common factor is 1. Y-intercept: the y-coordinate of the point at which a graph crosses the y-axis (where x = 0). X-intercept: the x-coordinate of the point at which it crosses at the x-axis (where y = 0). Section 2.3: Slope Slope: (rate of change) of a line is the ratio of change in y-coordinate to the corresponding change in x-coordinates. The slope measures how steep a line is. Suppose a line passes through  EMBED Equation and EMBED Equation , look at the change in the y and x coordinates:  EMBED Equation ,  EMBED Equation  History of the origin of the symbol m for slope: One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is "monter." However, there is no evidence to make any such connection. In fact, Descartes, who was French, did not use m. Eves (1971) suggests "it just happened." The earliest known example of the symbol of m appearing in print is O'Brien (1844). Salmon (1960) subsequently used the symbols commonly employed today to give the slope-intercept form of a line 4 types of slope: 1. Positive slope goes up from left to right, m = positive number. 2. Negative slope goes down from left to right, m = negative number. 3. NO slope are HOrizOntal lines, x = some number. 4. Undefined slopes are from vertical lines, you cannot ski down a vertical slope, y = some number. Real-World Examples: Slant of the roof on houses, a ladder leaning against the wall, physical view of the ocean floor, houses in California- on hills, road going up a hill. Section 2.4: Writing Linear Equations (Omit Algebra II) Slope-intercept form: y = mx + b, m is the slope and b is the y-intercept. Point-slope form: if you are given the coordinates of two points on a line, you can use this equation of the line that passes through them.  EMBED Equation  where  EMBED Equation  and  EMBED Equation  are the coordinates of a point on the line and m is the slope. If the lines are parallel, the slope stays the same If the lines are perpendicular then the slopes are opposite reciprocals. Section 2.5: Modeling Real-World Data: Using Scatter Plots Scatter Plots: is a graph in which two sets of data are plotted as ordered pairs in a coordinate plane. Scatter plots are used to investigate a relationship between two quantities. Type in scatter plots in your calculator: STAT, EDIT, ENTER, list your x values in L1 and your y values in L2. 2nd, y = , ENTER, ON, ENTER, TYPE: ENTER (THE FIRST ONE), Xlist: L1; Ylist: L2, Enter, GRAPH, ZOOM, 9 Linear Correlation: A correlation is a relationship between two things, or variables.For example, it is generally true that the more junk food you eat, the fatter you get. This is a positive correlation -there is a strong relationship. A negative correlation means that there is a strong relationship between two variables, but in the opposite direction. For example, the older a second-hand car is, the lower the price.Zero correlation happens when there is little or no relationship between two things-when they don't affect each other. For example, the number of times you go to the movies in a year is probably not related to the number of brothers and sisters you have. Calculator: STAT, CALC, 4, ENTER, the r is the linear coefficient. If r is close to negative one then you have a negative correlation. (r  EMBED Equation -1) If r is close to one you have a positive correlation. (r  EMBED Equation 1) If r is close to 0 there is NO correlation. (r  EMBED Equation 0)  INCLUDEPICTURE "http://www.writefix.com/graphs/corr/image001.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.writefix.com/graphs/corr/image002.gif" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.writefix.com/graphs/corr/image003.gif" \* MERGEFORMATINET  Section 2.6: Special Functions Greatest Integer Function: EMBED Equation , consists of line segments, or rays. You always round down when you take the greatest integer. Ex. EMBED Equation ,  EMBED Equation ,  EMBED Equation  The xs will be in an inequality form:  EMBED Equation  where you have a closed circle on the left and an open circle on the right, this is the width of the inequality. The ys are the height from one segment to the next. Domain: All Real Numbers. To graph: MATH, NUM, 5, GRAPH Real life example: the cost of postage to mail a letter is a function of the weight of the letter. Absolute Value Function: EMBED Equation , this function is in the shape of a V.  EMBED Equation  moves the number up or down, if the number is positive, it moves the graph up if the number is negative it moves the graph down.  EMBED Equation  moves the function in the opposite direction, from the sign, left to right. If the number you are adding is negative you move it to the right, if the number you are adding is positive you move it to the left.  EMBED Equation  the number in front makes the graph squeeze together  EMBED Equation  a fraction stretches the graph out Domain: All Real Numbers. To graph: MATH, NUM, 1, GRAPH. Piecewise Function: the absolute value function can be written a: f(x) = -x if x < 0 Graph it in two parts x if x  EMBED Equation  0 y = type in fxn, /(x < or > #) this part is the if part the graph. The graphs of each part of a piecewise function may or may not connect. A graph may stop at a given x value and then begin again at a different y value for the same x value. *The graph does not give you open or closed circles. 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