# دليل المعلم Functions From A Calculus Perspective الرياضيات الصف الثاني عشر

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دليل المعلم Functions From A Calculus Perspective الرياضيات الصف الثاني عشر

Common Error part c may confuse with wmt to many Remind that x is the variabe and that a. b. C and d are constants- Error Analysis For Exercise 60, students should use the three steps for determining continuity The "due f(c) does exist (Step t)- Ho•wever. the value of is different as c is from the left and c is approached from the right is correct frat the graph of the relatjon does not pass the vertical line test There are two y-values when x is c. The graph does regesent a function

4 Asses

Yesterday's News Ask students to describe how analyzing graphs of relatjons and functions helped them continuity and end behavior

A"tional

George; sample answer. The relation in the graph is not a function because there are two y -values paired with the same x - value

#### I Focus

Vertical Alignment

Before Lesson 1- 4 Find function values

Lesson 1- 4 Determine intervals on
which functions are increasing, constant. or decreasing. and determine maxima and mi ima of functions

Determine the average rate of change of a function

After Lesson 1- 4 Graph and describe parent functions

#### 2 Teach

Scaffolding Questions

Have students reu:i the Why ? sectiM of the lesson

• A business owner improves manufacturing processes after a steady drop in profit. The changes take place over June, July. and August. When should the graph of profit change from decreasing to
Sample answer. during or shortly after June, July, or August

• A senates monthly approval ratings have up and down over the last year. How could she find the average rate of change over two months ? She could subtroct the first month's rating from the second rnonth's rating md divide by 2-

1 Increasing and Decreasing Behavior

Example 1 shows how to fhd intervals on which a function is decreasing, or constant Examples 2—4 show how to find and use extrema. Formative Assessment Use the Guided Practice exercises after
each exatnple to students' understanding of concepts

Use the graph of function to estimate intervals to the nearest 0.5 unit on which the function is increasing, decreasing, or constant. Support the answer numerically-

ECONOMY Advertsen •Mts for a new car that a tank of gas will take a driver and three passengers 360 miles. After researchng on the Interræt. you find the function for miles tank of gas for the car is (x) = —0025x2 + 3.5x + 240. where x is the of ttp car in miles tur. What speed o—zes the distance the car can tavel on a tank of gas? How far wil the cu travel at that optinum speed

There is a mamum at 70 miles hour. The cu vall travel 362.5 miles when traveling at the optimum speed-

2 Average Rate of Change

Exarnples S and 6 show how to find average rates of change

Focus on Mathematical Content Rate of Finding the average rate of between two points of a function is the same as determining the of a Ihe- However. the rate of change two ponts of a nonlinear function can with every pair of When

calculating the of a line, think

When average rate of

chmge of a functbn, think

#### Follow - up

Students have explored characteristics of functions

• How does understanding parent functions and transformations help you to represent mathematical ideas and analyze real-world situations

Sample answer: Understanding the relationship between parent functions allows you to choose an appropriate function that could be used to represent a real - world situation

What characteristics of functions can yaj analyze red-wuld situations ? Explain. Sample answer- End behavior recyesents future behavior, critkd pints represent maximum and min*num values', rates of change represent speeds and other changes

#### 1 Focus

Vertical Alignment

Before Lesson 1-5 Analyze graphs of

Lesson IS kientify. graph, and parent functions-nklentify and gra# trmsformatjons of parent functions

After Lessm 1- 5 Perform operations With and find compositions of functions

2 Teach

Scaffolding Ouestions

Have students read the Why ? section of the lessm

• What are tt•e similarities md differences between (x) = x and glx) = x + 2 ? The slope lines is same. g(x) shifted up 2 LÄits • how values of c will affect the gaph of f(x) = x + a The value of c shifts the line up or down I o I units. ntjnued on ttæ next

#### 1 Focus

Usea graphing calculator to solve nonlinear inequalities

Teaching Tip

stucknts turn on their calculatMS. explain the first step the Activity. The first inequality iS derived the right Side of the inequality with y. The second inequality iS derived by replacing the left side of the inequality With y

2 Teach

Working in Cooperative Groups Pair students who are with a graphing calculator With who are not. Have exh pair take turns entering information into the calculator

Pracitke Have students complete Exercises 1.3.5.7. and 8

3 Assess

Formative Assessment

Have students work independently to Exercises 2, 4. and 6. Have students draw their onto a sheet of paper. along with solution sets

From Concrete to

Ask to summarize how to find the solution region of two or nonlinear inequalities

#### I Focus

Vertical Alignment

Before Lesson 1- 6 Evaluate functiMs

Lesson 1- 6 Perform operations with Find compositions of

After Lesson 1- 6 Find inverses of relations and functions

2 Teach

Scaffolding Questions

Have students read the Why ? sectim of the lesson- • Depending on the month, two products sell at different rates. How can we compare the rates ? Subtract the sales for one product from the sales for the other

• A comp.any is interested in the monthly ratio of homes sold to homes for sale. What express.iM rqyesents this ratio ? Divide the homes sold by the hornes for sale

#### WatchOut

Error Analysis In 73

interpretations are Ask to verify this by first translating the parent function 4 units up and then translMing the parent function 4 units to the right. They will see the result of steps is the same