WEEK 12-EVALUATING-LINEAR-FUNCTIONS

Objectives

Start

Evaluating Linear Functions

1

2

Story

Linear Functions

Checking for Linearity

Summary

Some Characteristics

Story

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Linear Functions

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What is a Linear Function?

Definition: A linear function is in the form:

Here:

are constants.

mand b

Example 1: is a linear function,

Where:

f(x)=mx+b

y=2x-4

m=2 b=-4

x,y

are the variables

is the independent variable.

x

We are not interested in knowing the meaning of m and b right now. That is a matter for the next lesson.

Linear Functions can be visually represented on a graph.

Example 2: Graph function defined on all of the real numbers.

f(x)=3x-1

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1. Select two (*) values of x you want and replace them into the function. (*) It's enough to select two values because only one unique line passes through two points.

2. f(0) = 3(0)-1=-1

3. f(1) = 3(1)-1=2

1. Plot each point obtained in the table as (x,y)

2. Join points with a line. You can extend it as long as domain of function is all real numbers.

Some Characteristics of Linear Functions

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🗨️

Linear functions exhibit a constant rate of change. This means that for every increase in x, there is a proportional increase or decrease in y.

1

Example: Linear Function increases 2 units in y for every increment of 1 in x.

y=2x+4

↷↷↷↷

↺↺↺↺

+2 +2 +2 +2

+1 +1 +1 +1

1. Select some values of x with the same increment/raise.

2. Evaluate these values on the function: f(-2) = 2(-2)+4 = 0 f(-1) = 2(-1)+4 = 2 f(0) = 2(0)+4 = 4 ...

3. Find the difference between each of the values of y.

Taking Decisions at Supermarket

Maria is planning to buy some oranges and found the following table of prices in the supermarket:

a) Determine how much increases the price for each additional orange b) Plot the points on a cartesian plane.

Solution

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Prices in Colombian pesos.

Checking for Linearity

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Let's see some examples:

Example 2: Determine whether the values of the table belong to a linear function.

Example 3: Determine if the function is a linear function or not. Graph it.

y=-x+3

↷ ↷ ↷ ↷

↷ ↷ ↷ ↷

↺ ↺ ↺ ↺

↺ ↺ ↺ ↺

-1 -1 -1 -1

+1 +3 +5 +7

+1 +1 +1 +1

+1 +1 +1 +1

It is not a linear function

It is a linear function

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It is enough to note that y=-x+3 is in the same format as f(x)=mx+b. Then, by inspection, it clearly is a linear function.

3. If y increased by the same amount, then the function would be linear, but y increases differently for each value of y.

4. The function increases the same amount, then the function is linear.

1. Find the increment of x.

3. Find the difference between each of the values of y.

1. Select some values of x with the same increment.

3. Find the difference between each of the values of y.

07:00

Try it by Yourself

You are a financial analyst and need to determine if a savings account can be modeled as a linear function base on the deposits per month made by a client.

a) Does given data follow a linear function? c) Plot points on a Cartesian Plane and if they are linear join them with a line.

🤔

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Prices in Peruvian Soles.

Link the type of function with the set of points.

Linear

Nonlinear

Linear

Linear

{(-1,30),(1,15),(3,0)}

{(0,0),(1,1),(2,2)}

{(0,-500),(0,-400),(0,-300)}

{(0,0),(1,1),(2,8)}

• {(-1,30),(1,15),(3,0)} -> Linear
• {(0,0),(1,1),(2,2)} -> Linear
• {(0,-500),(0,-400),(0,-300)} -> Linear
• {(0,0),(1,1),(2,8)} -> Nonlinear

The solution will appear in 45 seconds

Click here to draw >

Try it by yourself:

LINEAR

NONLINEAR

GRAPHIC

RAISE (SLOPE)

Summary:Functions

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Constant rate of change of linear functions. Example: In f(x)=2x+4 x-2-1012y02468 Rate of change=+2

A straight line.

Not a straight line.

Welcome 6th graders!

A journey soon begin through Social Science experiences!

Great job!

See you next time

8TH-EVALUATING-LINEAR-FUNCTIONS-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0

It is highly advised to have:

MATERIAL

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A compass.
• A Protactor.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).

Solution

Step 3

Step 2

Step 1

Answer

Step 4

b)

a)

Deposits follow a linear pattern/function.

↷ ↷ ↷ ↷

↺ ↺ ↺ ↺

+169 +169 +169 +169

+1 +1 +1 +1

1. Find how each value of x raises.

3. How price raises is a characteristic of the linear function, in this case: $550 for each additional orange.

3. Find the difference between each of the values of y.

1. Plot each point in the table as (x,y)

2. Join points with a line. You can extend it as long as domain of function is all real numbers.

MA.8.F.1 Define, evaluate and compare functions.MA.8.F.1.2 "Given a function defined by a graph or an equation, determine whether the function is a linear function. Given an input-output table, determine whether itcould represent a linear function."MA.K12.MTR.5.1 Use patterns and structure to help understand and connect mathematical concepts.

Solution

Step 3

Step 2

Step 1

Answer

Step 4

b)

a)

Orange prices raise $550 for each additional unit.

↷ ↷ ↷ ↷

↺ ↺ ↺ ↺

+550 +550 +550 +550

+1 +1 +1 +1

1. Find how each value of x raises.

3. How price raises is a characteristic of the linear function, in this case: $550 for each additional orange.

3. Find the difference between each of the values of y.

1. Plot each point in the table as (x,y)

2. Join points with a line. You can extend it as long as domain of function is all real numbers.