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CHAPTER 5: GRAPHS AND FUNCTIONS 5-1 RELATING GRAPHS TO EVENTS You can use an equation, an inequality, or a proportion to make a statement about a variable. You can use a graph to show the relationship between two variables. 5-2 RELATIONS AND FUNCTIONS A RELATION is a set of ordered pairs. The (age, height) ordered pairs below form a relation. IDENTIFIYING RELATIONS AND FUNCTIONS Giraffes Age (years) 18 20 21 14 18 Height(meters) 4.25 4.40 5.25 5.00 4.85 You can list the set of ordered pairs in a relation using braces. {(18, 4.25), (20, 4.40), (21, 5.25), (14,5.00), (18, 4.85)} Recall that a function is a relation that assigns exactly one output (range) value for each input (domain) value. One way you can tell whether a relation is a function is to analyze the graph of the relation using the vertical line test. If any vertical line passes through more than one point on the graph, then for some value of x there is more than one value of y. Therefore the relation is not a function. EVALUATING FUNCTIONS You can think of a function as an input- output machine. If you know the input values you can use the function rule to find the output values. The output values depend on the input values. y = 3x + 4 INPUT OUTPUT X Y 1 7 2 10 3 13 Another way to write the function y = 3x + 4 is f(x) = 3x + 4 A function is in FUNCTION NOTATION when you use f(x) to indicate the outputs. Example: Evaluate the function rule f(x) = -3x + 5 to find the range of functions for the domain {-3,1,4} 5-3 FUNCTION RULES, TABLES, AND GRAPHS CONTINUOUS DATA – Data where numbers between any two data values have meaning. Use a solid line to indicate continuous data. DISCRETE DATA: Data that involve a count of items, such as the number of people, or the number of cars. For discrete data indicate each item with a point. Graph the function: y = |x| + 1 Graph the function f(x) = x^2 + 1 5-4 WRITING A FUNCTION RULE You can write a rule for a function by analyzing a table of values. Look for a pattern relating the independent and dependent variables. Write a function rule for the table: x f(x) 1 -1 2 0 3 1 4 2 Answer: f(x) = x – 2 x y 1 2 2 4 3 6 4 8 Answer: y = 2x x y 1 3 2 4 3 5 4 6 Answer: y = x + 2 5-5 DIRECT VARIATION DIRECT VARIATION – A function in the form y = kx, where k does not equal 0, is a direct variation. The constant of variation for direct variation k is the coefficient x. The variables y and x are said to vary directly with each other. WRITING AN EQUATION OF DIRECT VARIATION Is the equation a direct variation? If it is find the constant of variation. Example: 5x + 2y = 0 2y = -5x y = -5/2x The equation is in the form of y = kx, so the equation is a direct variation and the constant of variation is -5/2. Example: 5x + 2y = 9 2y = -5x + 9 y = -5/2x + 9/2 The equation can not be written in the form of y = kx. Write an equation given a point. Write an equation of direct variation given the point (4, -3) y = kx -3 = k(4) -3/4 = k y = -3/4x PROPORTIONS AND EQUATIONS OF DIRECT VARIATIONS For the table, use the ratio y/x to tell whether y varies directly with x. x y y/x -3 2.25 2.25/-3 = -.75 1 -.75 -.75/1 = -.75 4 -3 -3/4 = -.75 6 -4.5 -4.5/6 = -.75 5-6 INVERSE VARIATION INVERSE VARIATION: An equation in the form xy = k or y=k/x where k does not equal 0, is an INVERSE VARIATION. The CONSTANT OF VARIATION FOR INVERSE VARIATION is k, the product of x * y for an ordered pair (x,y). Inverse variations have graphs with the same general shape. The smaller the constant of variation, the closer the curve is to the axis. Write an equation given a point. Suppose y varies inversely with x and y = 7 when x = 5. Write an equation for the inverse variation. xy = k 5(7) = k 35 = k xy = 35 Find the missing coordinate. The points (3,8) and (2,y) are two points on the graph of an inverse variation. Find the missing value. x1 * y1 = x2 * y2 3(8) = 2(y2) 24 = 2(y2) 12 = y2 COMPARING DIRECT AND INVERSE VARIATION Does the data in each table represent direct or inverse variation? x y 2 5 4 10 10 25 x y 5 20 10 10 25 4 COPY THE GRAPHS FOR DIRECT AND INVERSE VARIATIONS. 5-7 DESCRIBING NUMBER PATTERNS INDUCTIVE REASONING AND NUMBER PATTERNS Inductive reasoning is making conclusions based on patterns you observe. A conclusion you reach by inductive reasoning is a conjecture. WRITING RULES FOR ARITHMETIC SEQUENCE A number pattern is also called a sequence. One kind of number sequence is an arithmetic sequence. You form an arithmetic sequence by adding a fixed number to each previous term. This fixed number is a common difference. For example: -4, 5, 14, 23 +9 +9 +9 9 is the common difference. Try… 11,23,35,47, …. 8,3,-2,-7, …. Consider the sequence 7, 11, 15, 19, …. Think of each term as the output of a function. Think of the term number as the input. Term number 1 2 3 4 input Term 7 11 15 19 output You can use the common difference of the terms of an arithmetic sequence to write a function rule for the sequence. For this sequence the common difference is 4. Let n = the term number of the sequence. Let A(n) = the value of the nth term of the sequence. A(1) = 7 A(2) = 7 + 4 = 7 + 1*4 A(3)= 7 + 4 + 4 = 7 + 2 * 4 A(4)= 7 + 4 + 4 + 4 = 7 + 3 * 4 A(n) = 7 + 4 + 4 + 4 + …. + 4 = 7 + (n - 1)*4 RULE: A(n) = a + (n-1) d nth first term common term term number difference