Let's Get Started

We're going to analyze various situations to understand what the slope and y-intercept represent.  You will watch videos to further your understanding.  Before you get started, you may want to print out the worksheet, "What's Slope Got to Do With It," by clicking here so you can work on it on your own paper.

TEKS Standards and Student Expectations

A(3) Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:

A(3)(A) determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and

A(3)(B) calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems

Resource Objective(s)

Given algebraic, tabular, graphical, or verbal representations of linear functions in problem situations, the student will determine the meaning of slope and intercepts as they relate to the situations.

Essential Questions

What does the slope represent?

What does the y-intercept represent?

Vocabulary

Real World Examples and Slope-Intercept Form: Video

The video below uses a real-world situation to explain the meaning of slope and the y-intercept. Watch the video, and answer the following questions.

Source: 
Linear Equations – Real Life Situation Solved in Slope-Intercept Form, hannahmnations, YouTube

Rental Car Example

 

It costs $32 per day to rent a medium-sized car, plus $0.25 for each mile driven over 150 miles. The equation that represents this situation is y = 0.25x + 32, where represents the total cost of renting a car, and x represents the number of miles driven over 150 miles.

Look at the graph of y = 0.25x + 32 that is shown at the right and answer the related questions. 

 

If you look at the labels on the axes, you will see that the meaning of slope can be found by writing the labels in the   formula.

 = .

The cost per mile is $0.25.

Post Office Example

The cost to send a package at the post office is $0.33 for the first ounce, plus $0.22 for each additional ounce. The equation that represents this situation is y = 0.22x + 0.33, where y represents the cost of mailing the package and represents the cost for each additional ounce over 1 ounce.

Let’s look at a table of data that represents this situation.

Weight over 1 ounce, x (in ounces) Total Cost, y (in dollars)
0 0.33
1 0.55
2 0.77
3 0.99
4 1.21

 

If you look at the column headings of the table, you will see that the meaning of slope can be found by writing the column headings in the   formula.

=  

 

The cost per ounce is $0.22

Look at the graph of y = 0.22x + 0.33 that is shown below and answer the related questions.

 

Car Trip Example

A car being used for a trip to the Geyser starts with a full tank of 15 gallons of gasoline. The car gets 25 miles per gallon.

The equation that represents this situation is y = -x + 15, where y represents the gasoline in the gas tank, and x represents the miles used per gallon of gasoline.

Look at the graph of y = -x + 15  that is shown below and answer the related questions.

If you look at the labels on the axes, you will see that the meaning of slope can be found by writing the labels in the   formula.

  =   The gallons used per mile is .

Writing an Equation in Slope-Intercept Form From a Table of Values

What's Slope Got to Do With It?

Investigating Slope and y-Intercept in the Real World

Directions: Use what you have learned about the concepts of slope and y-intercept to solve the following four problems. Enter your answers in the Journal Activity below. Alternatively, you can print the problems by clicking here.


A. A Day at the Fair

You and your friends plan to attend the annual county fair this weekend. The entry fee for the
carnival is $5.00, and the cost per ticket is $0.50.

Number of Tickets Cost
8 $9.00
12 $11.00
  $12.50
23  

1. Complete the above table.

2. Write a linear equation in which y represents the total cost, and x represents the number of rides selected.

3. Identify the slope and y-intercept in the equation, and explain what each of them represents in the context of the problem.

     slope (m) = 

     y-intercept = 

4. Sketch the graph of the linear equation.

5. Your parents have decided to give you $20.00 to spend at the fair. If you need three
tickets for each ride, how many rides will you be able to go on? Use mathematics to
explain your answer. Use words, symbols, or both. (You may write your answers in the Journal Activity below.)

B. Canoe Caper

You and your brother decide to go boating while your family is visiting Deep Creek Lake. The
park rangers require a $25.00 deposit to rent a canoe, and a rental fee of $6.50 per hour.

1. Write a linear equation in which y represents the total cost of renting a canoe and x represents the number of hours spent on the canoe. ______________________

2.  Identify the slope and y-intercept in the equation, and explain what each of them represents within the context of the problem.

     Slope (m) =

     y-intercept (b) =

3. Graph the linear equation, using your graphing calculator. Sketch it below on the coordinate grid.

4. Using the TRACE or table function on the calculator, find the cost of renting a canoe from 12:30 to 3:30 p.m., provided that the canoe is returned in the same condition in which you received it.

     Cost:

Use mathematics to explain how you got your answer. Use words, symbols, or both. (You may write your answers in the Journal Activity.)

C. Sweet Sixteen

Your parents have decided to buy an SUV for $25,635 and they have promised that the SUV will be yours when the car is worth $10,000. According to the car dealer, the SUV will depreciate in value approximately $3,000 per year.

1.  Write a linear equation in which y represents the total value of the car and x represents
the age of the car.


2.  Identify the slope and y-intercept in the equation and explain what each of them
represents within the context of the problem.

     m =

     b =

3. Graph the linear equation using your graphing calculator. Sketch the line below on the coordinate grid.

4. a. Based on the information above, will the SUV be yours on your sixteenth birthday if your parents bought it when you were twelve years old?

    b.  If not, how old you will be when the SUV is finally yours?

Explain your answer by using mathematics. Use words, symbols, or both. (You may write your answers in the Journal Activity.)

D. Can We Talk?

You have just signed an annual contract for a cellular phone. The base rate is $32 per month for 200 minutes, and $0.14 per minute for all additional minutes.

1. Write a system of linear equations in which y represents the total cost for the cell phone per month and x represents the number of minutes spent on the phone each month.

      First 200 minutes:

      More than 200 minutes:

2. Identify the slope (m) and y-intercept (b) in the linear equation, which represents the cost of the cellular phone when using more than 200 minutes in a given month. Then explain what the slope and y-intercept represent within the context of the problem.

     m =

     b =

3. Examine the linear equation which represents the cost of the cellular phone when using fewer than 200 minutes. Why is it different from the other equation? Justify your answer by using mathematics. Use words, symbols, or both. (You may write your answers in the Journal Activity.)

4. Graph the two linear equations in your graphing calculator and sketch the results on the coordinate grid.

5. Your cell phone bill for last month was $629.40, and you know that this cost is much too high. If you talked on your cellular phone for 221 minutes last month, how much do you believe your bill should have been? Explain your answer using mathematics. Use words, symbols, or both. (You may write your answers in the Journal Activity.)

Kid2Kid Video

Rachelle is a student tutor helping a student understand the meaning of slope and intercepts. Watch the video to see Rachelle's suggestions.

Writing Equations to Describe Functional Relationships (English Version)
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Determinando el Significado de la Pendiente y Puntos de Intersección (Spanish Version)
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Source: 
https://www.youtube.com/watch?v=jByeP3mrx6Y

Vocabulary Activity

Journal Activity