Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

M3: Section 12.8, 8.1-8.3 Notes Page 1 of 20 Academic Chapter 8 Notes Linear Functions Sections 12.8, 8.1-8.3 Name_____________________Pd.___ M3: Section 12.8, 8.1-8.3 Notes Sections 12.8, 8.1-8.3 List of Vocabulary Words: Section 12.8: Sequence Term Arithmetic sequence Common difference Geometric sequence Common ratio Section 8.1: relation domain range input output function vertical line test Section 8.2: equation in two variables solution of an equation in two variables graph linear equation function form Section 8.3: x-intercept y-intercept Page 2 of 20 M3: Section 12.8, 8.1-8.3 Notes Page 3 of 20 Section 12.8: Sequences Learning Goal: We will extend and graph sequences. Vocabulary: Sequence – Term – Arithmetic sequence – a sequence in which the difference between consecutive terms is constant Common difference – the difference between consecutive terms of an arithmetic sequence EXAMPLE 1: Extending Arithmetic Sequences Find the common difference for the arithmetic sequence. Then find the next three terms. a. 32, 39, 46, 53, 60,… b. 88, 83, 78, 73, 68,… M3: Section 12.8, 8.1-8.3 Notes ON YOUR OWN: c. 12, 17, 22, 27, 32,… Page 4 of 20 d. 5, 3, 1, -1, -3, … Geometric sequence – a sequence in which the ratio of any term to the previous term is constant Common ratio – the ratio of any term of a geometric sequence to the previous term of the sequence EXAMPLE 2: Extending Geometric Sequences Find the common ratio for the geometric sequence. Then find the next three terms. a. 24, 48, 96, 192, … b. 625, 125, 25, 5, … c. 2, 6, 18, 54, … d. 160, 80, 40, 20, … M3: Section 12.8, 8.1-8.3 Notes Page 5 of 20 ON YOUR OWN: EXAMPLE 3: Using Sequences You are saving for $1100 laptop. In August, you save $380 from a summer job. Starting in September, you save $120 per month from a part time job. After how many months of saving will you have enough money for the laptop? ON YOUR OWN: M3: Section 12.8, 8.1-8.3 Notes EXAMPLE 4: Graphing an Arithmetic Sequence Graph the arithmetic sequence 25, 21, 17, 13, 9, … Graph the arithmetic sequence 5, 10, 15, 20, 25, … EXAMPLE 5: Graphing a Geometric Sequence Graph the sequence 20, 30, 45, 67.5, 101.25, …. Page 6 of 20 M3: Section 12.8, 8.1-8.3 Notes Page 7 of 20 Section 8.1: Relations and Functions Learning Goal: We will use graphs to represent relations and functions. Vocabulary: Relation – a pairing of numbers in one set (the domain) with the numbers in another set (range) Domain – Range – Input – Output – EXAMPLE 1: Identifying the Domain and Range Identify the domain and range of the relation represented by the table below, which shows one alligator’s length at different ages. M3: Section 12.8, 8.1-8.3 Notes Page 8 of 20 ON YOUR OWN: Identify the domain and range of the relation. **In addition to using ordered pairs or a table to represent a relation, you can also use a ______________ or a _____________________. EXAMPLE 2: Representing a Relation Represent the relation (2, 0), (1, -1), (2, 2), (0, 0), (-1, 1) as indicated. a. A graph. b. A mapping diagram ON YOUR OWN: Represent the relation (-1, 1), (2, 0), (3, 1), (3, 2), (4, 5), as indicated. a. A graph. b. A mapping diagram M3: Section 12.8, 8.1-8.3 Notes Page 9 of 20 Function – a relation with the property that for each input there is exactly one output EXAMPLE 3: Identifying Functions Tell whether the relation is a function using a mapping diagram. Explain your reasoning. a. (1, 60), (2, 100), (3, 160), b. (2, 0), (1, 1), (2, 2), (0, 0), (4, 210), (5, 270) (-1, 1) ON YOUR OWN: Tell whether the relation is a function. Explain your reasoning. a. (0, 3), (1, 2), (2, -1), (4, 4), b. (-2, -1), (0, 2), (2, 3), (-2, -4) (5, 4) Vertical line test – for a relation represented by a graph, if any vertical line passes through more than one point of the graph, then the relation is not a function. If no vertical line passes through more than one point of the graph, then the relation is a function. M3: Section 12.8, 8.1-8.3 Notes Page 10 of 20 EXAMPLE 4: Using the Vertical Line Test Tell whether the relation represented by the graph is a function. a. b. M3: Section 12.8, 8.1-8.3 Notes Page 11 of 20 Section 8.2: Linear Equations in Two Variables Learning Goal: We will find solutions of equations in two variables. Vocabulary: Equation in two variables – an equation that contains two different variables Solution – an ordered pair (x, y) that produces a true statement when the coordinates of the ordered pair are substituted for the variables in the equation EXAMPLE 1: Checking Solutions Tell whether the ordered pair is a solution of a. (4, 2) ON YOUR OWN: b. (2, 1) x 3 y 1. M3: Section 12.8, 8.1-8.3 Notes Page 12 of 20 EXAMPLE 2: Finding Solutions The number of pages p that Donald has left to write for his 30-page research paper depends on how many days d he writes 5 pages per day. This situation can be modeled by the equation p 30 5d . a. Make a table of solutions for the equation. b. How many days does he need to finish the paper? ON YOUR OWN: M3: Section 12.8, 8.1-8.3 Notes Page 13 of 20 Graph – the set of points in a coordinate plane that represent all the solutions of the equation Linear equation – an equation whose graph is a line EXAMPLE 3: Graphing a Linear Equation Graph y 1 x 1. 2 ON YOUR OWN: Graph the equation. a. y x 3 b. y 3x 4 M3: Section 12.8, 8.1-8.3 Notes **The graph of the equation y Page 14 of 20 b is the _________________ line through (0, b). **The graph of the equation x a is the _________________ line through (a, 0). EXAMPLE 4: Graphing Horizontal and Vertical Lines Graph the equation. a. x 1 b. y 4 ON YOUR OWN: Graph y 1 and x 4 . Tell whether each equation is in function form. M3: Section 12.8, 8.1-8.3 Notes Page 15 of 20 Function form – an equation that is solved for y EXAMPLE 5: Writing an Equation in Function Form a. Write 5 x y 8 in function b. Write 3 x 2 y 6 in function form. form. c. Write x 3 y 3 in function form. Then graph the equation using a t-chart. M3: Section 12.8, 8.1-8.3 Notes Page 16 of 20 ON YOUR OWN: 1 a. Write x y 3 in function form. Then graph the equation using 2 a t-chart. b. Write 2 x 3 y 3 in function form. Then graph the equation using a t-chart. M3: Section 12.8, 8.1-8.3 Notes Page 17 of 20 Section 8.3: Using Intercepts Learning Goal: We will use x- and y-intercepts to graph linear equations. **You can graph a linear equation quickly by recognizing that only ____ points are needed to draw a line. Vocabulary: x-intercept – the x-coordinate of a point where a graph crosses the x-axis y-intercept – the y-coordinate of a point where a graph crosses the y-axis M3: Section 12.8, 8.1-8.3 Notes Page 18 of 20 EXAMPLE 1: Finding Intercepts of a Graph Find the intercepts of the graph of 2y x 2 . ON YOUR OWN: Find the intercepts of the equation’s graph. 1. x 2 y 2 2. 4 x 3 y 12 3. y 2 x 8 EXAMPLE 2: Using Intercepts to Graph a Linear Equation Graph the equation 2y x 2 . (Use the intercepts from Example 1) M3: Section 12.8, 8.1-8.3 Notes Page 19 of 20 ON YOUR OWN: Find the intercepts of the equation’s graph. Then graph the equation. 1. x 2 y 2 (On Your Own #1) 2. 4 x 3 y 12 (On Your Own #2) EXAMPLE 3: Writing and Graphing an Equation While at the beach, you can rent a boogie board for $1 per hour or a surfboard for $2 per hour. You have $10 to spend. Write and graph an equation describing the possible combinations of hours x renting a boogie board and hours y renting a surfboard that you can afford. M3: Section 12.8, 8.1-8.3 Notes ON YOUR OWN: Page 20 of 20