Grade 10 - Math - LO.6 - Modeling With Functions: Linear And Step Functions

  

We have in LO.6 Math G10

First: the Concepts

-Linear functions

-Piecewise function

-Absolute-Value Functions

-STEP function

-Direct variation

-Inverse variation

-Intercepted part

-Composition of functions

-Translation of linear function

Second: the References

-Math Connections 1A 3.4 - 3.6, 4.4 - 4.5

-Math Connections 1B 6.1 - 6.4, 6.6

Third: the Videos links

Fourth: Skills

-Given data, create functions that model the applied situations. Calculate a least-squares regression line, understand its properties, compute and interpret correlation for data

-Interpret the functions based on the real-world situation they model

-Analyze the functions and their characteristics. Explain the different properties of the algebraic function and its graph.

-Identify, interpret and describe real-world step functions

-Analyze direct and inverse variations based on the real-world situation they model.

-Graph linear, piecewise, absolute value and step functions

-Find the inverse of some functions algebraically and graphically and determine if the inverse is a function. (Also: Distinguish between "inverse variation" and "inverse function.")

-Solve real-life application on functions

Fifth: the materials as PPT., DOCX., and PDF

In the Drive from this link

Few Notes:

What is a Function?

A function relates an input to an output.

It is like a machine that has an input and an output.

And the output is related somehow to the input.

f(x)	"f(x) = ... " is the classic way of writing a function. And there are other ways, as you will see!

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

• The input
• The relationship
• The output

Example: "Multiply by 2" is a very simple function.

Here are the three parts:

Input	Relationship	Output
0	× 2	0
1	× 2	2
7	× 2	14
10	× 2	20
...	...	...

For an input of 50, what is the output?

Some Examples of Functions

x² (squaring) is a function
x³+1 is also a function
Sine, Cosine and Tangent are functions used in trigonometry
and there are lots more!

But we are not going to look at specific functions ...
... instead we will look at the general idea of a function.

Names

First, it is useful to give a function a name.

The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want.

But let's use "f":

We say "f of x equals x squared"

what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows us the function is called "f", and "x" goes in

And we usually see what a function does with the input:

f(x) = x² shows us that function "f" takes "x" and squares it.

 

Example: with f(x) = x²:

• an input of 4
• becomes an output of 16.

In fact we can write f(4) = 16.

 

The "x" is Just a Place-Holder!

Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

f(x) = 1 - x + x²

Is the same function as:

• f(q) = 1 - q + q²
• h(A) = 1 - A + A²
• w(θ) = 1 - θ + θ²

The variable (x, q, A, etc) is just there so we know where to put the values:

f(2) = 1 - 2 + 2² = 3

 

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

y = x²

But there is still:

• an input (x)
• a relationship (squaring)
• and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it!

A function relates an input to an output.

Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

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