Warm UP Write down objective and homework in agenda Lay out homework (Real world Graphs worksheet) Homework (Interpreting Worksheet & Review Sheet)

Unit 3 Functions -Common Core Standards 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Note: At this level, focus on linear and exponential equations F-BF.1 Write a function that describes a relationship between two quantities. 1)Determine an explicit expression, a recursive process, or steps for calculation from a context. 2) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Note: At this level, limit to addition or subtraction of constant to linear, exponential or quadratic functions or addition of linear functions to linear or quadratic functions. F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Even and odd functions are not addressed. F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Unit 3 Functions -Common Core Standards F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: At this level, the focus is linear and exponential functions. F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Note: At this level, focus on linear, exponential and quadratic functions; no end behavior or periodicity. F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Note: At this level, focus on linear and exponential functions F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: At this level, focus on linear functions and exponential functions whose domain is the subset of integers. N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Warm UP

Important Vocabulary! Independent variable: the variable that provides the input for a function. It is the variable that stands by itself and determines what the outcomes will be. Dependent variable: the variable that is the output of a function. It is determined by a value of the independent variable.

Interpreting Graphs In this lesson you will continue to You will also describe graphs using the words increasing, decreasing, linear, and nonlinear match graphs with descriptions of real-world situations You will also use intervals of the domain to help you describe a function’s behavior write a description of a real-world relationship displayed in a graph draw a graph to match a description of a real-world situation

Investigation: Matching Up A function is linear if, as x changes at a constant rate, the function values change at a constant rate. The graphs of linear functions appear as straight lines. A function is nonlinear if, as x changes at a constant rate, the function values change at a varying rate. The graphs of non-linear functions are curved.

EXAMPLE Use the intervals marked on the x-axis in the graph below to help you discuss where the function is increasing or decreasing and where it is linear or nonlinear. The first interval is done for you. Interval 1: The function is decreasing in interval 1 & 2. In interval 1, the function is nonlinear and decreases slowly at first and then more quickly. Intervals 2-4??

Example A turtle crawls steadily from its pond across the lawn. Then a small dog picks up the turtle and runs with across the lawn. The dog slows down and finally drops the turtle. The turtle rests for a few minutes after this excitement. Then a young girl comes along, picks up the turtle, and slowly carries it back to the pond.

A turtle crawls steadily from its pond across the lawn A turtle crawls steadily from its pond across the lawn. Then a small dog picks up the turtle and runs with across the lawn. The dog slows down and finally drops the turtle. The turtle rests for a few minutes after this excitement. Then a young girl comes along, picks up the turtle, and slowly carries it back to the pond. Which of the graphs depicts the turtle’s distance from the pond over time? Explain.

Select one of the other three graphs Select one of the other three graphs. Work with your partner to write a story that would be depicted by the graph. (You can use the turtle or another situation.)

Story Before a volleyball game starts, the people that can be found in the school gym are the players, coaches, and the people working the event (ticket takers, officials, scorers, etc.) Slowly the fans arrive for the match. Just before the first game, the people are coming in as fast as the tickets can be sold. After the match is over, most of the parents and fans leave. Then more students arrive for the after-game dance. Most of the students leave after an hour. The people that remain are the ones who have been working at the gym all night long.

Before a volleyball game starts, the people that can be found in the school gym are the players, coaches, and the people working the event (ticket takers, officials, scorers, etc.) Slowly the fans arrive for the match. Just before the first game, the people are coming in as fast as the tickets can be sold. After the match is over, most of the parents and fans leave. Then more students arrive for the after-game dance. Most of the students leave after an hour. The people that remain are the ones who have been working at the gym all night long. What is the independent variable for this situation? What are reasonable values for the domain? Are they positive or negative numbers? Whole numbers or decimals? What is the dependent variable? Sketch a graph that matches the story. Be sure to label the axes. Is your graph discrete or continuous? Explain why you drew it that way.  Compare your graph to your partner’s graph. How are they alike? How are they different? Are both graphs reasonable?

Stories to Graphs Example 1: Josephine walked 40 feet. At the halfway point, she had walked for 25 seconds. She stopped for 5 seconds to tie her shoe and then continued walking for 25 more seconds.

Stories to Graphs Example 2 You are mowing the lawn. As you mow, the amount of grass to be cut decreases. You mow at the same rate until about half the grass has been cut. Then you take a break for a while. Then, mowing at the same rate as before, you finish cutting the grass. Sketch a graph that shows how much uncut grass is left as you mow, take your break, and finish mowing.

Stories to Graphs Example 3 This graph represents a flag being raised on a flagpole. Write a story that describes what is happening to the flag and explain the shape of the graph.

The person raising the flag raises it a short distance, rests and regrips the cord and then raises it more. This continues until it is fully raised. By the length of the diagonal part, it is raised a little less each interval as it gets higher and the person gets tired.

A man walks from his home to the subway stop A man walks from his home to the subway stop. He waits for the train to arrive. He then takes a subway to commute to work. The subway makes one stop. Make a graph below to show what is described and label the parts.

Sketch a graph of the altitude of a pelican, who takes off from shore, circles to search for fish, and dives to the water to catch a fish. Label each section.