Course Name: Math I Unit # 5 Unit Title: Functions and their Graphs Part II
BY THE END OF THIS UNIT:
Course Name: Math I Unit # 5 Unit Title: Functions and their Graphs Part II
CORE CONTENT
Cluster Title: Interpret functions that arise in applications in terms of a context.Standard: 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.
Concepts and Skills to Master:
· Given a graph, identify key features such as x-and y-intercepts; intervals where the function is increasing, decreasing, positive or negative; maxima or minima; symmetry and end-behavior
· Given a table of values, identify key features such as x- and y-intercepts; intervals where the function is increasing, decreasing, positive or negative; maxima or minima; symmetry and end-behavior
· Find key features of a function and use them to graph the function
· Use interval notation and symbols of inequality to communicate key features of graphs
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Ability to graph a quadratic, absolute value or piecewise-defined function from a table or equation
Academic Vocabulary
Increasing, decreasing, positive, negative, intervals, intercepts, interval notation, maximum, minimum
Suggested Instructional Strategies:
· Use graphing technology to explore and identify key features of a function.
· Use key features of a function to graph functions by hand
NCPDI Unpacking:
F-IF.4 This standard should be revisited with every function your class is studying. Students should be able to move fluidly between graphs, tables, words, and symbols and understand the connections between the different representations. For example,
when given a table and graph of a function that models a real-life situation, explain how the table relates to the graph and vice versa. Also explain the meaning of the characteristics of the graph and table in the context of the problem as follows:
At the course one level, the focus is on linear, exponentials, and quadratics
• Linear – x/y-intercepts and slope as increasing/decreasing at a constant rate.
• Exponential- y-intercept and increasing at an increasing rate or decreasing at a decreasing rate.
• Quadratics – x-intercepts/zeroes, y-intercepts, vertex, intervals of increase/decrease, the effects of the coefficient of x2 on the concavity of the graph, symmetry of a parabola. / Resources:
· Textbook Correlation: 4-2, 4-3, 5-3, 5-4, 5-5, 7-6, 7-7, 9-1, 9-2, 9-7, 11-7
Sample Assessment Tasks
Skill-based task
Identify the intervals where the function is increasing and decreasing.
/ Problem Task
Create a story that would generate a quadratic, absolute value or piecewise-defined function and describe the meaning of key features of the graph as they relate to the story.
Ex. Below is a table that represents the relationship between daily income, I, for an amusement park and the number of paying visitors in thousands, n.
n / I
0 / 0
1 / 5
2 / 8
3 / 9
4 / 8
5 / 5
6 / 0
a. What are the x-intercepts and y-intercepts and explain them in the context of the problem?
b. Identify any maximums or minimums and explain their meaning in the context of the problem.
c. What pattern of change do these ordered pairs develop? Explain.
d. Is the pattern and/or graph of the data symmetrical? How do you know?
e. Describe the intervals of increase and decrease and explain them in the context of the problem.
CORE CONTENT
Cluster Title: Interpret functions that arise in applications in terms of a context.Standard: 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.
Concepts and Skills to Master:
· Identify domains of functions given a graph
· Graph a function given a restricted domain
· Identify reasonability of a domain in a particular context
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Familiarity with function notation and domain
· Knowledge of independent and dependent variables
Academic Vocabulary
Domain, function, integers, independent variables, dependent variable, restricted domain
Suggested Instructional Strategies:
· Discuss contexts where the domain of a function should be limited to a subset of integers, positive or negative values, or some other restriction to the real numbers
· Find examples of functions with limited domains from other curricular areas (science, physical education, social studies, consumer science…)
· Focus on quadratic functions and compare with linear and exponential functions studied earlier in the course.
NCDPI Unpacking:
F-IF.5 Given a function, determine its domain. Describe the connections between the domain and the graph of the function. Know that the domain taken out of context is a
theoretical domain and that the practical domain of a function is found based on a contextual situation given, and is the input values that make sense to the constraints of the problem context. / Resources:
· Textbook Correlation: 4-4, 7-6, 9-1, 11-6
Sample Assessment Tasks
Skill-based task
You are hoping to make a profit on the school play and have determined the function describing the profit to be f(t) = 8t – 2654 where t is the number of tickets sold. What is a reasonable domain for this function? Explain. / Problem Task
Create functions in context where the domain would be:
a. All real numbers
b. Integers
c. Negative integers
d. Rational numbers
e. (10, 40)
Ex. A rocket is launched from 180 feet above the ground at time t = 0. The function
that models this situation is given by
h(t) = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet.
a. What is the theoretical domain for the function? How do you know this?
b. What is the practical domain for t in this context? Explain.
c. What is the height of the rocket two seconds after it was launched?
d. What is the maximum value of the function and what does it mean in context?
e. When is the rocket 100 feet above the ground?
f. When is the rocket 250 feet above the ground?
g. Why are there two answers to part e but only one practical answer for part d?
h. What are the intercepts of this function? What do they mean in the context of this problem?
i. What are the intervals of increase and decrease on the practical domain? What do they mean in the context of the problem?
CORE CONTENT
Cluster Title: Interpret functions that arise in applications in terms of a context.Standard: 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.
Concepts and Skills to Master:
· Calculate rate of change given a linear function from the equation or a table
· Calculate rate of change over a given interval in an exponential function from an equation or a table where the domain is a subset of the integers
· Use a graph to estimate the rate of change over an interval in a linear or exponential function.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Definition of slope
Academic Vocabulary
· Increasing, decreasing, rate of change, average, function, interval
Suggested Instructional Strategies:
· Use graphical data from birthrates, BMI in growing children, electricity rates, population growth or other linear or exponential data to explore and discuss the meaning of rate of change
NCDPI Unpacking:
F-IF.6 Students should be able to describe patterns of changes from tables and/or graphs of linear and exponential functions. Sample vocabulary may include, “increasing/decreasing at a constant rate” or “increasing or decreasing at an increasing or decreasing rate.” Students should be comfortable in their understanding of rates of change to apply their knowledge linear and non-linear graphical display. / Resources:
· Textbook Correlation: 5-1, CC-14
· Drug filtering
· Growth Rate
Sample Assessment Tasks
Skill-based task
Find the average rate of change on the interval [-3, 1]
Table1
X / -3 / -2 / -1 / 0 / 1 / 2 / 3
Y / 8 / 3 / -2 / -7 / -12 / -17 / -22
Table 2
X / -3 / -2 / -1 / 0 / 1 / 2 / 3
Y / 6 / 12 / 24 / 48 / 96 / 192 / 384
Table 3
X / -3 / -2 / -1 / 0 / 1 / 2 / 3
Y / 7 / 2 / -1 / 0 / 2 / 4 / 6
/ Problem Task
The graph models the speed of a car. Tell a story using the graph to describe what is happening in various intervals.
Ex. Kierra is a saleswoman whose pay depends on the number of diamonds she sells. She earns a base salary plus a commission on each sale. Using the table below, determine the rate of change in earnings as sales increase. What part of Kierra’s pay does this represent?
Number of Diamonds / 2 / 4 / 6 / 8
Weekly Earnings / 600 / 960 / 1320 / 1680
Ex. The graph below shows the population of Leicester from
1950 to 1990.
Use the graph to answer the following questions.
a. What was the average rate of change from 1975 to 1985?
b. What does this rate of change mean in the context of the problem?
c. Note- this could be used as a standalone question without parts a. and b. If the population increased by the same average rate from the year 1990 to 2005 as it did from 1975 to 1985, approximately what is the expected population in the year 2005? Justify your answer.
CORE CONTENT
Cluster Title: Analyze functions using different representationsStandard:
F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Concepts and Skills to Master:
· Graph parabolas expressed in vertex form or factored form by hand.
· Graph piecewise-defined functions by hand
· Use technology to model complex square root and piecewise-defined functions
· Identify intercepts, maxima, minima, end behavior, increasing/decreasing intervals, and axis of symmetry in graphs of quadratic functions
· Identify intercepts, maxima, minima, end behavior, increasing/decreasing intervals, in graphs of piecewise-defined functions.
SUPPORTS FOR TEACHERS
Critical Background Knowledge· Graph points on the coordinate plane
Academic Vocabulary
· Quadratic, piecewise-defined, step function, intercept, end behavior, increasing, decreasing, maximum, minimum, axis of symmetry
Suggested Instructional Strategies:
· Allow students to develop graphs from tables and use those graphs to generalize graphing strategies.
· Graph equations generated from real-life contexts
NCDPI Unpacking:
F-IF.7 This standard should be seen as related to F-IF.4 with the key difference being students can create graphs, by hand and using technology, from the symbolic function in this standard. / Resources:
· Textbook Correlation: 9-1, 9-2, CC-6, CC-16, CB5-8, 10-5
· Geogebra (free online)
Sample Assessment Tasks
Skill-based task
1. Graph the function f(x) = 3|2x + 1| - 2 and identify any maxima or minima
2. Graph the function y = (x – 3)(x + 2), identify the axis of symmetry and the vertex / Problem Task
Jose created a distance vs. time graph by starting 2 meters away from the wall. He walked towards to wall at 0.25 m/sec for 4 seconds, stood still for 2 seconds, walked away from the wall at 0.4 m/sec for 2 seconds, and then stopped for 2 seconds. Sketch Jose’s distance vs. time graph. What was Jose’s ending position? Give both the distance from the wall and the time.
Ex. The all-star kicker kicks a field goal for the team and the path of the ball is modeled by
f (x) = - 4.9t2 + 20t
Find the realistic maximum and minimum values for the path of the ball and describe what each means in the context of this problem.
CORE CONTENT
Cluster Title: Analyze functions using different representationsStandard:
F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t; y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Concepts and Skills to Master:
· Factor quadratic equations and identify zeros
· Use properties of exponents to interpret expressions for exponential functions
SUPPORTS FOR TEACHERS