Exploring Linear Functions

Exploring Linear Functions

Period 1 & 2

Topics include

1. Understanding equations written in problems, or Graphs & Tables

2. Domain​ of a function

3. Solving for future values

​Lets begin with a warm up question for Topic #1

Multiple Choice

Lee charges $3 for a basket and $2.50 for each pound of fruit picked at the orchard. Write an equation for the total cost of x pounds of fruit from the orchard.

• y = 3x + 2.5
• y = 2.50x + 3
• y = 2.50x 
• y = 3x + 2.50

​Consider this question, & note the difference in the language.

​This is still a linear equation. But Let us begin by identifying the variables. Since they use the word equation, we must use x and y.

​

Poll

What variable would you use for x and y?

• x for students 

  and 

  y for non-students

• y for students

  x for non-students

So Far we should undertand the set up.

​So which equation does not model/match this situation above?

​We will work on this problem on the white board to see which looks similar.

1. ​First we will simplify as is and check.

2. ​Second we will re-arange into y=mx+b

3. ​Third we will re-arange and simplify.

Multiple Choice

Identify the form of the following linear equation:

5x + 3y = 15

• Slope - Intercept Form

• Standard Form

• Neither

Multiple Choice

Convert the standard form equation into slope-intercept form.
x + 5y = -25

• $$y=-\frac{1}{5}x\ -5$$  

• y = -x - 5
• x = 20
• y = 27, 239, 430

Multiple Choice

Convert the standard form equation into slope-intercept form.
5x - 4y = -20

• $$y\ =\ \frac{5}{4}x\ +\ 5$$  

• y = -5x - 4
• x = 20
• y = 27, 239, 430

Open Ended

This is known as a standard form of a linear equation 12x - 6y = 30

Convert this equation to slope intercept form.

Exploring Linear Functions​

​Topic #1 Part II

​Understanding Parts of an Equation

Vocabulary:

• ​Total

• ​Initial value

• ​Rate

• ​Slope

• ​Starting Point

• ​y-intercept

Understanding Parts of an Equation

Vocabulary:

• ​Total

• ​Initial value

• ​Rate

• ​Slope

• ​Starting Point

• ​y-intercept

​Let us answer a few questions about the parts

​to this equation given in the problem.

​

​We will use the vocabulary words learned in class.

Multiple Select

$$C=.1998s+76.450$$  

What part of the equation is 76.450

• Slope

• Initial Value

• Total

• Y-intercept

Multiple Select

$$C=.1998s+76.450$$  

What part of the equation is .1998

• Slope

• Initial Value

• Rate

• Y-intercept

Open Ended

$$C=.1998s+76.450$$  

What is the variable s referring to in the question? 

AND

Why is it connected to .1998?

Understanding Parts of an Equation

Vocabulary:

• ​Total

• ​Initial value

• ​Rate

• ​Slope

• ​Starting Point

• ​y-intercept

Now that we know the slope/rate is .1998

​we can correctly answer the question...

​Exploring Linear Equations

​Topic 1 Part 3

Open Ended

a) How will you find the slope from the table?

b) What rate of change do you think best fits?

Compare your results with a neighbor

• When working with a table or graph it is best to find the most common change in x and y

​

• ​We can then confidently say this is the best fit for the data.

In Summary

​Exploring Linear Equations

​Topic 2 and 3

• ​This last question comes in 3 parts.

• ​First we will understand the situation represented in the graph.

• ​Then we will go through our steps, understanding the equation and seeing where things fit.

Open Ended

Practice question #1

Using the graph provided, Identify the y intercept and slope for function e(t) and m(t).

Then write the equation for both.

Scenario

​The cost of a business is the amount of money spent to start the business and keep it in operation.

​

The revenue is the amount of money the business earns from the sale of goods or services.

​

An online business called Muffins A-Plenty started on January 1 of a recent year. The lines in the graph represent the cost function and the revenue function of the first six months that Muffins A-plenty has been in operation.

​First let's focus on the graph

Open Ended

Open Ended

Part A

Write equations to represent the cost function and the revenue function for this business.

Open Ended

Part B

The profit of a business equals revenue minus cost. Using your cost and revenue functions from Part A find the Profit.

Open Ended

Part C

Determine after how many months of operation Muffins A-plenty will make its first profit.

You can determine this mathematically or contextually.

Open Ended

Part D

Predict the profit after 10 months of operation. Round your answer to the nearest dollar.