Warm UP Write down objective and homework in agenda Lay out homework (ROC worksheet) Homework (Real World Graphs Worksheet)

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 Given the function: h(x) = -3x -5, find h(4) & h(-3). Given the function: f(x) = x2+ 3x - 4, find f(-1) & f(4)

Warm Up Given the function: h(x) = -3x -5, find h(4) & h(-3). Given the function: f(x) = x2+ 3x - 4, find f(-1) & f(4) F(-1)= -6 & F(4) = 24

Graphs of Real-World Situations In this lesson you will describe graphs using the words increasing, decreasing, linear, and nonlinear match graphs with descriptions of real-world situations learn about continuous and discrete functions use intervals of the domain to help you describe a function’s behavior

What is the initial depth of the water? 2 feet For what time interval(s) is the water level decreasing? What accounts for the decrease(s)? 2 hours to 6 hours & 10 hours to 16 hours There is a leak in the pool!

For what time interval(s) is the water level increasing For what time interval(s) is the water level increasing? What accounts for the increase(s)?  6 to 10 hours Someone is filling up the pool! Is the pool ever empty? How can you tell? No! The depth never drops below 1 foot

In this example, the depth of the water is a function of time. That is, the depth depends on how much time has passed. So, in this case, depth is called the dependent variable. Time is the independent variable. When you draw a graph, put the independent variable on the x-axis and put the dependent variable on the y-axis. Domain: All real numbers 0 ≤ x ≤ 16 Range: All real numbers 1 ≤ x ≤ 3.5

What is the independent variable? How is it measured? Time, seconds This graph shows the volume of air in a balloon as it changes over time What is the independent variable? How is it measured? Time, seconds What is the dependent variable? How is it measured? Volume, cubic inches For what intervals is the volume increasing? What accounts for the increases? 2 ≤ x ≤ 4, 5.5 ≤ x ≤ 8, 9 ≤ x ≤ 12 Someone is adding air to make the air balloon rise

Someone is decreasing the air to make the balloon sink This graph shows the volume of air in a balloon as it changes over time For what intervals is the volume decreasing? What accounts for the decreases? 8 ≤ x ≤ 9, 16 ≤ x ≤ 21 Someone is decreasing the air to make the balloon sink For what intervals is the volume constant? What accounts for this?  4 ≤ x ≤ 5.5, 12 ≤ x ≤ 16 No one is putting air in the balloon What is happening for the first 2 seconds? There is no air in the balloon

Read the description of each situation below Read the description of each situation below. Identify the independent and dependent variables. Then decide which of the graphs above match the situation. White Tiger Population A small group of endangered white tigers are brought to a special reserve. The group of tigers reproduces slowly at first, and then as more and more tigers mature, the population grows more quickly. Independent Variable: Dependent Variable: Matching Graph:   Temperature of Hot Tea Grandma pours a cup of hot tea into a tea cup. The temperature at first is very hot, but cools off quickly as the cup sits on the table. As the temperature of the tea approaches room temperature, it cools off more slowly.

Read the description of each situation below Read the description of each situation below. Identify the independent and dependent variables. Then decide which of the graphs above match the situation. White Tiger Population A small group of endangered white tigers are brought to a special reserve. The group of tigers reproduces slowly at first, and then as more and more tigers mature, the population grows more quickly. Independent Variable: Dependent Variable: Matching Graph:   Temperature of Hot Tea Grandma pours a cup of hot tea into a tea cup. The temperature at first is very hot, but cools off quickly as the cup sits on the table. As the temperature of the tea approaches room temperature, it cools off more slowly.

Read the description of each situation below Read the description of each situation below. Identify the independent and dependent variables. Then decide which of the graphs above match the situation. Number of Daylight Hours over a Year’s Time In January, the beginning of the year, we are in the middle of winter and the number of daylight hours is at its lowest point. Then the number of daylight hours increases slowly at first through the rest of winter and early spring. As summer approaches, the number of daylight hours increases more quickly, then levels off and reaches a maximum value, then decreases quickly, and then decreases more slowly into fall and early winter. Independent Variable: Dependent Variable: Matching Graph:   Height of a Person Above Ground Who is Riding a Ferris Wheel When a girl gets on a Ferris wheel, she is 10 feet above ground. As the Ferris wheel turns, she gets higher and higher until she reaches the top. Then she starts to descend until she reaches the bottom and starts going up again.

Make Your Own Story! Choose one of the graphs that you have not matched yet (or sketch your own) and create a real-world situation that would match the graph. Describe the situation below and identify the independent and dependent variables. Indicate which graph you chose Independent Variable: Dependent Variable: Graph: