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CHAPTER 8 Introduction to Functions SECTION 8-1 Equations in Two Variables DEFINITIONS Open sentences in two variables– equations and inequalities containing two variables EXAMPLES 9x + 2y = 15 y = x2 – 4 2x – y ≥ 6 DEFINITIONS Solution– is a pair of numbers (x, y) called an ordered pair. Example State whether each ordered pair is a solution of 4x + 3y = 10 (4, -2) (-2, 6) DEFINITIONS Solution set– is the set of all solutions satisfying the sentence. EXAMPLE Solve the equation: 9x + 2y = 15 if the domain of x is {-1,0,1,2} SOLUTION x -1 0 1 (15-9x)/2 [15-9(-1)]/2 [15-9(0)]/2 [15-9(1)]/2 y 12 15/2 3 Solution (-1,12) (0,15/2) (1,3) 2 [15-9(2)]/2 -3/2 (2,-3/2) SOLUTION the solution set is {(-1,12), (0, 15/2), (1,3), (2,-3/2)} EXAMPLE Roberto has $22. He buys some notebooks costing $2 each and some binders costing $5 each. If Roberto spends all $22 how many of each does he buy? SOLUTION n = number of notebooks b = number of binders (n and b must be whole numbers) 2n + 5b =22 n = (22-5b)/2 SOLUTION b 0 2 4 (22-5b)/2 [22-5(0)]/2 [22-5(2)]/2 [22-5(4)]/2 n 11 6 1 Solution (0,11) (2,6) (4,1) 6 [12-5(6)]/2 -4 Impossible SOLUTION the solution set is {(0,11), (2, 6), (4,1)} SECTION 8-2 Points, Lines and Their Graphs COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-axis (abscissa)- the horizontal number line Y-axis (ordinate) - the vertical number line ORIGIN - the point where the x-axis and y-axis cross DEFINITION ORDERED PAIR - a unique assignment of real numbers to a point in the coordinate plane consisting of one xcoordinate and one ycoordinate DEFINITION GRAPH – is the set of all points in the coordinate plane whose coordinates satisfy the open sentence. LINEAR EQUATION is an equation whose graph is a straight line. Standard Form Ax + By = C where A, B, C are real numbers with A and B not both zero. If A, B, C are integers, the equation is in standard form. Example Are these equations in standard form? 2x – 5y = 7 0.5x + 4y = 12 2 x y + 3y = 4 1/x + 3y = 1 Graph the following lines 2x – 3y = 6 x = -2 y=3 SECTION 8-3 The Slope of a Line Property A basic property of a straight line is that its slope is constant. SLOPE is the ratio of vertical change to the horizontal change. The variable m is used to represent slope. FORMULA FOR SLOPE m = change in y-coordinate change in x-coordinate m = rise run Or SLOPE OF A LINE m = y2 – y1 x2 – x1 Find the slope of the line that contains the given points. M(4, -6) and N(-2, 3) Find the slope of the line that contains the given points. M(-2, 3) and N(4, 8) HORIZONTAL LINE a horizontal line containing the point (a, b) is described by the equation y = b and has slope of 0 VERTICAL LINE a vertical line containing the point (c, d) is described by the equation x = c and has no slope SECTION 8-4 The Slope Intercept Form of a Linear Equation SLOPE-INTERCEPT FORM y = mx + b where m is the slope and b is the y -intercept Y-Intercept is the point where the line intersects the y axis. X-Intercept is the point where the line intersects the x -axis. Find the slope and y-intercept and use them to graph each equation 1. y = -3/4x + 6 2. 2x – 5y = 10 THEOREM Let L1 and L2 be two different lines, with slopes m1 and m2 respectively. 1. L1 and L2 are parallel if and only if m1=m2 THEOREM and 2. L1 and L2 are perpendicular if and only if m1m2 = -1 Find the slope of a line parallel to the line containing points M and N. M(-2, 5) and N(0, -1) Find the slope of a line perpendicular to the line containing points M and N. M(4, -1) and N(-5, -2) SECTION 8-5 Determining an Equation of a Line Write an equation of a line with the given y-intercept and slope m=3 b = 6 Remember: y=mx+b THEOREM Let P(x1,y1) be a point and m a real number. There is one line through P having slope m. An equation of the line is y – y1 = m (x – x1) Write an equation of a line with the given slope, passing through the given point. m = 1/2; (-8, 4) Write an equation of a line passing through the given points A(1, -3) B(3,2) SECTION 8-6 Function Defined by Equations MAPPING DIAGRAM A picture showing a correspondence between two sets MAPPING – the relationship between the elements of the domain and range FUNCTION A correspondence between two sets, D and R, that assigns to each member of D exactly one member of R. DOMAIN – the set of all possible x-coordinates RANGE – the set of all possible y-coordinates RANGE The set R of the function assigned to at least one member of D. SECTION 8-7 Function Defined by Equations FUNCTION A correspondence between two sets, D and R, that assigns to each member of D exactly one member of R. DOMAIN – the set of all possible x-coordinates RANGE – the set of all possible y-coordinates FUNCTIONAL NOTATION f (x) denotes the value of f at x ARROW NOTATION f:x x+3 Is read the function f that pairs x with x + 3 VALUES of a FUNCTION The members of its range. EXAMPLE 1 Given f : x→4x – x2 with domain D= {1,2,3,4,5} Find the range of f. Example 2 Given g:x 4 + 3x – with domain D={-1, 0, 1, 2} 2 x Find the range of g. SECTION 8-8 Linear and Quadratic Functions Linear Function Is a function f that can be defined by f(x) = mx + b Where x, m and b are real numbers. The graph of f is the graph of y = mx +b, a line with slope m and y-intercept b. FUNCTION is a relation in which different ordered pairs have different first coordinates. RELATION Is any set of ordered pairs. The set of first coordinates in the ordered pairs is the domain of the relation, and RELATION and the set of second coordinates is the range. VERTICAL LINE TEST a relation is a function if and only if no vertical line intersects its graph more than once. Constant Function If f(x) = mx + b and m = 0, then f(x) = b for all x and its graph is a horizontal line y = b Determine if Relation is a Function {2,1),(1,-2), (1,2)} {(x,y): x + y = 3} SECTION 8-9 Direct Variation SECTION 8-10 Inverse Variation END