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Section 5 – 1 Relating Graphs to Events • In this section you will mostly be reading and interpreting graphs • Things to look for/remember: A) Notice the units on each of the axes B) Read the graph from left to right C) Points are named by their coordinates: (x-coordinate, y-coordinate) D) The x-coordinate tells you how to move horizontally (left or right) • Things to look for/remember (cont’d): E) The y-coordinate tells you how to move vertically (up or down) F) The point where the x-axis meets the yaxis is called the origin • Open your book to page 236 and we will look at the examples together Section 5 – 2 Relations and Functions • A relation is a set of ordered pairs • The domain is all of the x-coordinates in a set of ordered pairs (all possible x-values) • The range is all of the y-coordinates in a set of ordered pairs (all possible y-values) • When you are given a set of ordered pairs, name the domain and range by the set of numbers in set brackets { } • A function is a relation that passes the vertical line test If you were to draw a graph, you could draw a vertical line anywhere on that graph and it would never hit two or more points at the same time (no two points have the same x-value) in order to be a function See example 2 on page 242 • If any two y-values have the same x-value, the relation is NOT a function • A function rule is just an equation • The domain in an equation is all of the input values (all of the possible x-values) • The range in an equation is all of the output values (all of the possible y-values) • f(x) is function notation (say “f of x”) • Treat y = 2x + 5 the same as f(x) = 2x + 5 • Ex1. Find the domain and range of {(-2,4), (3,5), (6,0), (-1,7)}. Is it a function? • Ex2. Find the range of the function rule f(x) = 3x – 6 when the domain is {-2, 0, 4}. • Read ex. 3 on pg. 242 (mapping diagram) Section 5-3 Function Rules, Tables, and Graphs • x is the independent variable (the one you are inputting values for in the equation) • y is the dependent variable (the values are getting out from the equation) • y is the dependent variable because it depends on what x value you use • To graph a relation, you can make a table of x and y values and plot those values • If none of the variables has an exponent other than 1, your graph will be a line • Linear graphs: y = mx + b b is the y-intercept (it tells you where to put a point on the y-axis) m is the slope (from that y-intercept it tells you how to move to make another point) • Slope = rise/run = vertical change/horizontal change • If a variable has an exponent other than 1 or absolute value symbols, you will have to make a chart and plot enough points to see the shape of the graph • Graph each of the following: • Ex1. y = -3x + 2 Ex2. y = ⅔x – 6 • Ex3. y = |x| + 2 Ex4. y = x² – 2 Section 5-4 Writing a Function Rule • To write a function rule from a chart you must determine what operation(s) you can perform on the x-coordinate to get the corresponding y-coordinate each and every time • Ex1. Write a function rule x 1 2 3 5 f(x) 3 6 9 15 • When writing a function rule from a word problem, you will have to analyze the given information and figure out how to put it all together • Good place to start: determine what is the independent variable and what is the dependent variable • Ex2. Charles charges $15 per hour for babysitting and a flat fee of $3 for bringing his own movie with him. Write a rule to describe his profit as a function of how many n = # of hours P(n) = total profit hours he works. P(n) = 15n + 3 Section 5 – 5 Direct Variation • Direct variation describes the relationship between two things • For the variation to be direct, as one goes up so must the other • i.e. as you work more hours, you earn more money (these two things are direct variation) • Direct variation is written in the form y = kx where k ≠ 0 (k is called the constant of variation) • With direct variation (y = kx), y is said to vary directly with x (and vice versa) • To determine whether or not an equation is direct variation, solve for y and see if it ends up in the form y = kx If it does, then it is direct variation If it does not (i.e. y = ⅜x – 5), then it is not • Ex1. Write an equation of the direct variation that includes the point (-2, 6) y = kx 6 = k(-2) k = -3 y = -3x • Open your book to page 263 and read example 4 • Ex2. Do the “Check Understanding” on page 264 • Proportions are often used in direct variation questions (see example 5 on page 264) Section 5 – 6 Describing Number Patterns • Inductive reasoning is when you make conclusions based on patterns you have observed • A conclusion you reach is conjecture, because you are not sure whether or not it is true yet (it is just an educated guess) • You can use inductive reasoning to figure out number patterns (called sequences) • Each number in a sequence is called a term • Arithmetic sequences are those in which each term is determined by adding or subtracting the same number to the previous term (i.e. 3, 5, 7, 9, 11, …) • They are said to have a common difference • Arithmetic sequence A(n) = a + (n – 1)d A(n) = nth term of the sequence a = first term (many books write a1) n = term number d = common difference • Ex1. Find the 4th, 7th, and 10th terms of the sequence A(n) = 3 + (n – 1)(-5) • Ex2. Find the next three terms in the sequence and the constant difference: 3, -5, -13, -21,… . Then find the formula for the sequence.