Chapter 5 Graphs and Functions

Section 1: Relating Graphs to Events Graphs have rules to follow: ▫Read all graphs from LEFT to RIGHT ▫Pay attention to the units on each axis ▫Pay attention to the values on each axis ▫Points are named by their coordinates (x-coordinate, y-coordinate) ▫The x-axis is the horizontal axis ▫The y-axis is the vertical axis ▫Where the axes meet = the origin

A piecewise graph or piecewise function is one in which the graph can be cut into pieces ▫I▫It changes direction, shape, and/or slope at least one time on the graph Ex1. Sketch a graph of the altitude of a pelican, from take off from shore to diving in to the water to catch a fish. Label each section Open your book to page 236, we are going to look at the examples and try the 1 st “check understanding”

Section 2: Relations and Functions A relation is any set of ordered pairs ▫It can be a collection of points that are connected or not connected, as long as it can be graphed as one or more points Any number that is an x-coordinate is a part of the domain (write in set brackets) Any number that is a y-coordinate is a part of the range (write in set brackets) Ex1. Find the domain & range: {(-3, 6), (1, 1), (4, 9), (8, -2), (3, 5)}

To test whether or not a relation is a function ▫1▫1) draw a vertical line through the graph TThis is known as the vertical line test IIf the line touches 2 or more points on the graph it is NOT a function 2) look at the points in a chart If two or more of the x-values are the same it is NOT a function One way to write a function rule is by using function notation y = 2x + 3 becomes f(x) = 2x + 3 Ex2. Given f(x) = x² + 3, find f(-2) and f(5)

Ex3. Is the following relation a function? Justify your answer. {(-3, 4), (5, 7), (1, 9), (8, 2), (5, 3), (-1, 6)} Ex4. Find the range of the function rule y = 8x + 4 for the domain {-3, 6, 9}

Section 3: Function Rules, Tables and Graphs The first variable (usually x) is the independent variable ▫It is the one that you are plugging in to the equation The second variable (usually y) is the dependent variable It depends on what you plug in for x Make a table of values before making any graph (include a sufficient number of points)

If the only exponent is 1, the graph will be linear and you need 3 points in your table If it is any graph you will need more points (typically 5) so that you can truly see the shape of the graph Linear graphs: y = mx + b ▫m is the slope and b is the y-intercept If x is squared, then it will be a parabola If x is inside absolute value, then it will be v- shaped

The parabola will be up-side-down if the x² term is negative The v-shaped graph will be up-side-down if there is a negative in front of the absolute value Be sure to label your axes and x- and y-values Make a table of values and graph each of the following Ex1. Ex2. Ex3.

Section 4: Writing a Function Rule When writing a function rule from a table of values, you must find the pattern that can be applied to each x-value to get each y-value Ex1. Write a function rule for the table When writing a function rule from a word problem, read the question carefully, identify the key information, and determine how to write that information as an equation x y

Ex2. The journalism class makes $25 per page of advertising in the yearbook. If the class sells p pages of advertising how much money will it earn? Write a function rule to describe this relationship. Be sure to identify your variables.

Section 5: Direct Variation Two variables related in such a way that their values always have a constant ratio directly vary (i.e. direct variation) Equation for direct variation is y = kx where k ≠ 0 and k is known as the constant of variation You can say that x and y vary directly with each other when y = kx Ex1. Is 3x – 5y = 0 direct variation? If it is, find the constant of variation

Ex2. Write an equation of the direct variation that includes the point (-4, 10). When you are given a table of values you can determine whether the variables vary directly by dividing each y-value by its corresponding x-value ▫If the values are all the same, then it is direct variation ▫The value is the constant of variation (k) ▫See example 4 on page 263 Direct variation uses proportions (pg. 264)

For the data in each table, tell whether y varies directly with x. If it does, write an equation for the direct variation. Ex3. Ex4.

Section 6: Describing Number Patterns When you make an educated hypothesis based on things you have observed, you are using inductive reasoning and your conclusion is called a conjecture Inductive reasoning is used when figuring out numbers in a given sequence Each number in a sequence is a term The sequence is arithmetic if you can add or subtract the same number each time to get the next number in the sequence

That number is the common difference Arithmetic sequence: A(n) = a + (n – 1)d ▫A(n) is the nth term in the sequence ▫a is the first term in the sequence ▫n is the term position ▫d is the constant difference Ex1. Find the next 3 numbers in the sequence: 25, 21, 17, 13, … Ex2. Write the rule for the sequence in Ex1. Ex3. Find the 4 th and 10 th terms of the following sequence: A(n) = 6 + (n – 1)(7)