1. Five-Minute Check (over Lesson 2–2) Mathematical Practices Then/Now
New Vocabulary
Example 1: End Behavior of Linear Functions
Example 2: End Behavior of Nonlinear Functions
Example 3: Zeros and Extrema of a Graph
Example 4: Real-World Example: Find End Behavior and Extrema
Lesson Menu

2. State whether f(x) = 2 + x2 is linear.
A. yes
B. No, the variable has an exponent of 2.
5-Minute Check 1

3. State whether x – y = –6 is linear.
A. yes
B. No, none of the variables have exponents.
5-Minute Check 2

4. neither line nor point symmetry
Does the graph have line symmetry or point symmetry? If so, identify the line of symmetry or point of symmetry.
line symmetry; x = 0
line symmetry; x = 2
point symmetry; (0, 2)
neither line nor point symmetry
5-Minute Check 3

5. neither line nor point symmetry
Does the graph have line symmetry or point symmetry? If so, identify the line of symmetry or point of symmetry.
line symmetry; x = –3
line symmetry; (–3, 0)
point symmetry; (–3, 4)
neither line nor point symmetry
5-Minute Check 4

6. Mathematical Practices 1 Make sense of problems.
2 Reason abstractly and quantitatively.
4 Model with mathematics.
7 Make use of structure. MP

7. Content Standards
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.
F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. MP

8. Extrema and End Behavior
Section 2.3

9. You graphed continuous functions.
Identify the end behavior of graphs.
Identify extrema of functions.
Then/Now

10. The behavior of a graph as x approaches positive or negative infinity
end behavior
The behavior of a graph as x approaches positive or negative infinity
relative maximum
A point on the graph of a function where no other nearby points have a greater y-coordinate.
relative minimum
A point on the graph of a function where no other nearby points have a lesser y-coordinate
Vocabulary

11. The relative minima, relative maxima, and turning points
The points where a curve completely changes direction from either down to up or up to down.
Extrema
The relative minima, relative maxima, and turning points
Vocabulary

12. Describe the end behavior of each linear function.
End Behavior of Linear Functions
Describe the end behavior of each linear function. A.
Example 1A

13. End Behavior of Linear Functions
As x → +∞, f(x) → +∞ As the value of x approaches positive infinity the value of y will continue to increase and the graph will continue to increase and approach positive infinity.
As x → –∞, f(x) → –∞ As the value of x approaches negative infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity.
Example 1A

14. Answer: As x → +∞, f(x) → +∞, as x → –∞, f(x) → –∞.
End Behavior of Linear Functions
Answer: As x → +∞, f(x) → +∞, as x → –∞, f(x) → –∞.
Example 1A

15. Describe the end behavior of each linear function.
End Behavior of Linear Functions
Describe the end behavior of each linear function. B.
Example 1B

16. Answer: As x → +∞, g(x) → – 1.5, as x → –∞, g(x) → –1.5.
End Behavior of Linear Functions
As x → +∞, g(x) → –1.5 As the value of x approaches positive infinity the value of y will continue to be –1.5.
As x → –∞, g(x) → –1.5 As the value of x approaches negative infinity the value of y will continue to be –1.5.
Answer: As x → +∞, g(x) → – 1.5, as x → –∞, g(x) → –1.5.
Example 1B

17. Describe the end behavior of each nonlinear function.
End Behavior of Nonlinear Functions
Describe the end behavior of each nonlinear function. A.
Example 2A

18. End Behavior of Nonlinear Functions
As x → +∞, f(x) → +∞ As the value of x approaches positive infinity the value of y will continue to increase and the graph will continue to increase and approach positive infinity.
As x → –∞, f(x) → –∞ As the value of x approaches negative infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity.
Example 2A

19. Answer: As x → +∞, f(x) → +∞, as x → –∞, f(x) → –∞.
End Behavior of Nonlinear Functions
Answer: As x → +∞, f(x) → +∞, as x → –∞, f(x) → –∞.
Example 2A

20. Describe the end behavior of each nonlinear function.
End Behavior of Nonlinear Functions
Describe the end behavior of each nonlinear function. B.
Example 2B

21. End Behavior of Nonlinear Functions
As x → +∞, f(x) → –∞ As the value of x approaches positive infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity.
As x → –∞, f(x) → –∞ As the value of x approaches negative infinity the value of y will continue to decrease and the graph will continue to decrease and approach negative infinity.
Example 2B

22. Answer: As x → +∞, f(x) → –∞, as x → –∞, f(x) → –∞.
End Behavior of Nonlinear Functions
Answer: As x → +∞, f(x) → –∞, as x → –∞, f(x) → –∞.
Example 2B

23. Zeros and Extrema of a Graph
The table and graph below are of a function with extrema. Estimate the Zeros. Then estimate the coordinates at which relative maxima and minima occur.
Example 3

24. Zeros and Extrema of a Graph
Zeros at x = –2 and x = 0 The zeros of a function are where the graph crosses the x axis. The graph crosses the x axis at –2 and 0.
Relative minimum near x = –1 The relative minimum is the point where the curve on the graph is the lowest. The minimum is at x = –1.
Example 3

25. Zeros and Extrema of a Graph
Relative maximum near x = 0 The relative maximum is the point where the curve on the graph is the highest. The maximum is at x = 0.
Example 3

26. Zeros and Extrema of a Graph
Answer: zeros at x = –2 and at x = 0; relative minimum near x = – 1; relative maximum near x = 0
Example 3

27. Find End Behavior and Extrema
FINANCES The table and graph represent the balance in Fredrica's savings account over a year. Use the table and graph to estimate the extrema for this function. Then explain the extrema in the context of the situation.
Real-World Example 4

28. Relative maxima are at x = 5; 8 The maxima occur in May and August.
Find End Behavior and Extrema
Relative minima are at x = 6; 9 The minima occur in June and September.
Relative maxima are at x = 5; 8 The maxima occur in May and August.
The balance in the account is the lowest at the minima.
The balance in the account is the highest at the maxima.
Real-World Example 4

29. Find End Behavior and Extrema
Answer: Relative minima in June and September, or at x = 6, 9; relative maxima in May and August, or at x = 5, 8; Because June and September are relative minima, those are the months where the account balance is the lowest. Because May and August are relative maxima, those are the months where the account balance is the highest.
Real-World Example 4