Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after...

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Module 3.3 Modeling With Functions What is function notation, and how can you use functions to model real-world situations? P. 115

Transcript of Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after...

Page 1: Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after working 8 hours? P. 129 For these situations: 1) Identify the independent and dependent

Module 3.3

Modeling With Functions

What is function notation, and how can youuse functions to model real-world situations?

P. 115

Page 2: Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after working 8 hours? P. 129 For these situations: 1) Identify the independent and dependent

P. 127In the winter, more electricity is used when the outside temperature goes down, and less is used when the outside temperature rises.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

How do you know? _____________________________________________________

The cost of shipping a package is based on its weight.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

The faster Tom walks, the quicker he gets home.

The ___________________ depends on the ____________.

The dependent variable is ________________________. The independent variable is _________________.

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Write an algebraic EXPRESSION to represent this situation: Amanda charges $10 per hour for babysitting.

10x where x is the number of hours.

Write an algebraic EQUATION using 2 variables to represent this situation.

𝒚 = 𝟏𝟎𝒙

The ______________________ is dependent on the _______________________.

The dependent variable is ____. The independent variable is ____.

This is an example of a function, because for every input (𝒙), there’s only one output (𝒚).

When we represent a function in FUNCTION NOTATION, we replace 𝒚 with 𝐟(𝐱), which is pronounced “f of x”.

So in FUNCTION NOTATION, that function/equation above is written as 𝒇(𝒙) = 𝟏𝟎𝒙, and pronounced “f of x equals 10x”.

Any variable will work for the independent variable. So the above could also be written as 𝒇(𝒉) = 𝟏𝟎𝒉, 𝒇(𝒕) = 𝟏𝟎𝒕, etc.

P. 128

Page 4: Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after working 8 hours? P. 129 For these situations: 1) Identify the independent and dependent

P. 129For this situation:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

Function Notation

Once you’ve written a function rule,you can evaluate it for any value

of the independent variableby substituting that value into the rule

and simplifying.

You do this all the time in your head!

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Function Notation

P. 129For this situation:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

Page 6: Module 3.3 Modeling With Functions...Kate earns $7.50 per hour. How much money will she earn after working 8 hours? P. 129 For these situations: 1) Identify the independent and dependent

Stan, a local delivery driver, is paid $3.50 per mile driven plus a daily amount of $75. On Monday, he’s assigned a route that’s 30 miles long. How much is he being paid for that day?

Kate earns $7.50 per hour. How much money will she earn after working 8 hours?

P. 129For these situations:1) Identify the independent and dependent variables2) Write an equation in function notation3) Use the equation to solve the problem.

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The domain is all possible x-values or inputs.The range is all possible y-values or outputs.

Once you create a function to describe a real-world situation , you must choose a reasonable domain (x-values) and range (y-values). But – not all numbers are reasonable choices. For example, the length of an object can’t be negative, and only whole numbers can represent a number of people.

P. 130

Write a function in function notation for the situation, then find a reasonable domain and range.

Manuel has already sold $20 worth of tickets to the school play. He has 4 tickets left to sell at $2.50 per ticket. What is the total amount collected from ticket sales?

𝒚 = $𝟐. 𝟓𝟎 ∙ 𝒙 + $𝟐𝟎

or 𝒚 = 𝟐. 𝟓𝒙 + 𝟐𝟎

And in Function Notation: 𝒇(𝒙) = 𝟐. 𝟓𝒙 + 𝟐𝟎

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How do you find all possible values of the range? Substitute all domain values into the function rule, one at a time, and evaluate.

When x = 0, it becomes 2.5(0) + 20 = 0 + 20 = 20. We write that as 𝒇 𝟎 = $𝟐𝟎When x = 1, it becomes 2.5(1) + 20 = 2.5 + 20 = 22.50. We write that as 𝒇 𝟏 = $𝟐𝟐. 𝟓𝟎When x = 2, it becomes 2.5(2) + 20 = 5 + 20 = 25. We write that as 𝒇 𝟐 = $𝟐𝟓. 𝟎𝟎When x = 3, it becomes 2.5(3) + 20 = 7.5 + 20 = 27.50. We write that as 𝒇 𝟑 = $𝟐𝟕. 𝟓𝟎When x = 4, it becomes 2.5(4) + 20 = 10 + 20 = 30. We write that as 𝒇 𝟒 = $𝟑𝟎. 𝟎𝟎

So the range is {$20, $22.50, $25, $27.50, $30}.

We said the domain is all possible x-values – OR INPUTS.So the input to the function doesn’t have to be an “x”; it could be any variable.Meaning, the function above could be written as 𝒇(𝒕) = 𝟐. 𝟓𝒕 + 𝟐𝟎, or 𝒇(𝒑) = 𝟐. 𝟓𝒑 + 𝟐𝟎, etc.

The domain is all possible x-values (or inputs). In this case, Manuel has only 4 tickets to sell, meaning possible values for x are 0, 1, 2, 3, and 4. So a reasonable domain is {0, 1, 2, 3, 4}.

P. 130Function Notation: 𝒇(𝒙) = 𝟐. 𝟓𝒙 + 𝟐𝟎

Don’t forget the zero!

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Write a function in function notation for the situation, then find a reasonable domain and range.P. 130

A telephone company charges 25 cents per minute for the first 5 minutes of a call, plus a 45 centconnection fee per call. What is the total cost (in dollars) of making a call?

Let m represent the number of minutes used.

Or: 𝒇 𝒎 = 𝟎. 𝟐𝟓𝒎 + 𝟎. 𝟒𝟓

The charges only occur if a call is made, so a reasonable domain is { _____________________ }

Substitute these values into the function above to find the range.

And you get: { _____________________________________________________ }

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Write a function in function notation for these situations, then find a reasonable domain and range.P. 130

The temperature early in the morning is 17 °C. The temperature increases by 2 °C for every hourfor the next 5 hours. What is the new temperature?

Flora earns $8.50 per hour proofreading advertisements at a local newspaper. She worksno more than 5 hours a day. What are her earnings?

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How would you write 𝒙 + 𝒚 = 𝟔 in function notation?First solve for y: Subtract x from both sides, so you’d get 𝒚 = 𝟔 − 𝒙 or 𝒚 = −𝒙 + 𝟔,then replace y with f(x). So the answer would be: 𝒇(𝒙) = −𝒙 + 𝟔

How would you write 𝟑𝒙 + 𝟐𝒚 = 𝟔 in function notation?

A function can be thought of as a machine that processes inputs in a particular wayand always produces the same output for a given input.For example, when you input water into an ice machine, the output is always ice cubes. Can you think of other real-world objects that have an input and an output?

Create a function such that 𝒇(𝟑) = 𝟏𝟐, and describe a real-world situation that the function could represent.

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