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

Fill in the Blanks

Complete the slope-intercept form of the equation.

The

is missing in the problem?

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.