Lesson Menu 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

Over Lesson 2–2 5-Minute Check 1 A.yes B.No, the variable has an exponent of 2. State whether f(x) = 2 + x 2 is linear. Then find f(4).

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

Over Lesson 2–2 5-Minute Check 3 A.line symmetry; x = 0 B.line symmetry; x = 2 C.point symmetry; (0, 2) D.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.

Over Lesson 2–2 5-Minute Check 4 A.line symmetry; x = –3 B.line symmetry; (–3, 0) C.point symmetry; (–3, 4) D.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.

Over Lesson 2–2 5-Minute Check 5 A.y = 10x + 2; m = 10 and b = 2 B.y = 10x – 2; m = 10 and b = –2 C.y = –10x + 2; m = –10 and b = 2 D.–y = 10x + 2; m = 10 and b = 2 Write the equation 2 – y = 10x in the form y = mx + b. Identify m and b.

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

MP 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.

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

Vocabulary end behavior relative maximum relative minimum turning points extrema

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

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 End Behavior of Linear Functions

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

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

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

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

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

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

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

Example 3 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 Zeros and Extrema of a Graph Zeros at x = –2 and x = 0The 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 = –1The relative minimum is the point where the curve on the graph is the lowest. The minimum is at x = –1.

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

Example 3 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

Real-World Example 4 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 Find End Behavior and Extrema Relative minima are at x = 6; 9The minima occur in June and September. Relative maxima are at x = 5; 8The 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 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. Find End Behavior and Extrema