Write and graph a linear function by examining a real-life scenario

Summary

This lesson explains how to write and graph a linear function using a real-life scenario involving taxi fares. It walks through the process of determining the cost of a taxi ride based on a flat rate and a per-mile charge, highlighting key concepts such as constant rates of change, slope-intercept form, and identifying slope and intercepts. The video demonstrates how to model the scenario with equations, tables, and graphs, reinforcing the understanding of linear relationships. It also addresses common misunderstandings and illustrates how different points on a graph can satisfy the linear equation.

Takeaways

• 😀 The cost of a taxi ride can be calculated using a linear function with a fixed rate and a flat fee.
• 🛣️ Superior Cab Company charges $0.80 per mile with a $2 flat rate, making a 4-mile ride cost $5.20.
• 📉 A linear relationship has a constant rate of change and can be represented using a table, graph, or equation.
• 🔢 Slope-intercept form (y = mx + b) helps represent linear functions, where m is the slope (rate of change) and b is the y-intercept (starting value).
• 📏 The slope represents how much the cost increases for each mile traveled, and the y-intercept is the initial charge when no miles are traveled.
• 🚖 Real-life scenarios, like taxi fares, demonstrate linear relationships where cost changes consistently with distance traveled.
• 💸 For example, a taxi company charging $1.50 per mile plus a flat fee can be modeled with an equation.
• 📐 Using known values like distance traveled and cost, you can substitute them into the equation to solve for missing variables, like the y-intercept.
• 📊 Tables can be used to calculate the total cost for different distances by plugging values into the linear equation.
• 📈 Graphs of these functions show straight lines, reinforcing the linear nature of the relationship between miles traveled and total cost.

Q & A

What is a linear relationship?

A linear relationship is a relationship between two variables with a constant rate of change. This can be observed in tables, graphs (as a straight line), and real-life scenarios.

How can you identify the slope in a linear relationship?

To find the slope in a linear relationship, you calculate the change in y-values divided by the change in x-values between two points on the line. The slope is the constant rate of change.

What does the y-intercept represent in a linear equation?

The y-intercept is the starting point of the line, where it crosses the y-axis. It represents the value of the dependent variable (y) when the independent variable (x) is 0.

What is the slope-intercept form of a linear equation?

The slope-intercept form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept.

How does the Superior Cab Company determine the cost of a ride?

The Superior Cab Company charges a flat rate of $2 plus 80 cents for every mile traveled. This follows a linear relationship where the total cost depends on the distance traveled.

How would you calculate the cost of a four-mile cab ride with Superior Cab Company?

To calculate the cost of a four-mile cab ride, you would multiply the distance (4 miles) by 80 cents per mile and then add the flat rate of $2. This gives a total cost of $5.20.

What common mistake is made when interpreting the information in word problems?

A common mistake is incorrectly identifying key numbers, such as assuming a given distance or cost is the starting point, when it's actually an ordered pair in the linear function.

How do you model a taxi fare scenario as a linear equation?

First, identify the constant rate (slope) and any given points (like distance and cost). Use the slope-intercept form (y = mx + b) to model the scenario, where the y-intercept is determined by solving the equation with known values.

What is the purpose of creating a table when analyzing a linear scenario?

A table helps organize data by showing how the dependent variable (total cost) changes with the independent variable (distance). It allows you to calculate the cost for different distances and clearly see the linear relationship.

How can you represent a taxi fare scenario graphically?

You can graph a taxi fare scenario by plotting distance (x-axis) against cost (y-axis). The result should be a straight line that reflects the constant rate of change, with the y-intercept representing the initial flat rate.

Outlines

00:00

🚖 Determining the Cost of a Taxi Cab Ride through Linear Relationships

In this section, the cost of a four-mile taxi ride is determined by examining a real-life scenario involving a linear function. The example introduces the Superior Cab Company, which charges $0.80 per mile and a flat rate of $2. The cost of the ride can be modeled through linear functions, where the relationship between distance traveled and cost is constant. The slope-intercept form, y = mx + b, is reviewed to understand how a linear relationship can be derived from tables, graphs, and real-life examples. The section also emphasizes the importance of correctly identifying key numbers, such as the rate per mile and total cost, to avoid common mistakes when solving word problems in linear functions.

05:01

🔢 Using Slope and Y-Intercept to Model Taxi Cab Scenarios

This part dives deeper into linear functions by modeling a taxi fare based on mileage. It explains the concept of constant rate (slope) and y-intercept, highlighting that the $1.50 per mile rate in a new scenario represents the slope, while the total cost for 3.5 miles ($7.25) provides valuable data for solving the equation. The explanation includes substituting values into the equation to determine the y-intercept, which turns out to be 2. By using this information, the final linear equation, c = 1.5d + 2, is developed, showing how distance and total cost relate through a consistent formula.

Keywords

💡Linear Function

A linear function represents a relationship between two variables with a constant rate of change. In the video, it's explained using a taxi fare scenario where the cost is a function of miles traveled, showcasing how a linear function models real-world situations with constant rates like taxi fares.

💡Slope-Intercept Form

The slope-intercept form is a standard way of writing linear equations, expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The video teaches how to derive this form from real-life data like the taxi fare, showing how the cost per mile represents the slope and the base fare represents the y-intercept.

💡Rate of Change

The rate of change is the slope in a linear relationship, indicating how much one variable changes with respect to another. In the video, the rate of change is represented by the cost per mile (e.g., $1.50 per mile), illustrating the concept of constant increase in taxi fare as the miles increase.

💡Y-Intercept

The y-intercept is the point where the line crosses the y-axis in a graph, representing the starting value when the independent variable is zero. In the taxi fare example, the y-intercept is the flat rate charged regardless of distance, which is $2 in the scenario provided.

💡Independent Variable

The independent variable is the variable that is manipulated or changed, typically plotted on the x-axis. In the taxi fare scenario, the independent variable is the distance traveled (in miles), since the fare depends on how far the passenger goes.

💡Dependent Variable

The dependent variable depends on the independent variable and is plotted on the y-axis. In the video, the total cost of the taxi ride is the dependent variable because it changes based on the number of miles traveled, illustrating the concept of dependent relationships.

💡Constant Rate

A constant rate refers to a fixed amount of change in one variable with respect to another, represented by the slope in a linear function. In the taxi fare scenario, the constant rate is $1.50 per mile, which remains the same for any distance traveled.

💡Ordered Pair

An ordered pair consists of two values that represent points on a graph, written as (x, y). In the video, the taxi ride data (distance, cost) provides ordered pairs like (3.5 miles, $7.25), which can be plotted to visualize the linear relationship between miles traveled and total fare.

💡Table of Values

A table of values lists corresponding pairs of independent and dependent variables to visualize how one affects the other. The video shows how to create a table of values for the taxi fare, illustrating the cost for different distances such as 0, 1, 2, and 3 miles, helping students understand the linear pattern.

💡Graph

A graph visually represents the relationship between two variables, where the slope and intercept determine the shape of the line. The video discusses how plotting the taxi fare data on a graph creates a straight line, confirming the linear relationship between miles traveled and total cost.

Highlights

A four-mile taxi ride with Superior Cab Company costs $5.20, based on a flat rate of $2 and a variable rate of $0.80 per mile.

A linear relationship between two variables involves a constant rate of change, often observed in tables or graphs as a straight line.

Slope-intercept form is expressed as y = mx + b, where m is the slope and b is the y-intercept.

To find the slope, calculate the change in y-values over the change in x-values, demonstrated as -3/6 resulting in a slope of -1/2.

The y-intercept (b) is the point where the line crosses the y-axis, determined by the x-value being 0.

In real-life scenarios, identifying the constant rate of change is crucial for solving linear equations, such as $0.80 per mile for a taxi ride.

Common mistakes include misinterpreting given values like distance traveled as starting points, when they should be used to form ordered pairs.

The equation for a taxi ride can be modeled in slope-intercept form, with the slope representing the cost per mile and the y-intercept as the flat fee.

A real-world example: A taxi ride costs $1.50 per mile and a total of $7.25 for 3.5 miles, with the cost and distance forming an ordered pair (3.5, 7.25).

The equation can be solved by substituting known values to find the y-intercept, resulting in a model like c = 1.5d + 2.

A table can be used to represent the cost for different distances, showing that the cost increases by $1.50 for each additional mile.

The equation, table, and graph all illustrate a linear relationship between distance traveled and total cost in a cab ride.

Graphing the relationship confirms the linearity, as the plotted points form a straight line.

The linear equation can be used to calculate costs for any fractional distances, such as half or quarter miles.

All points on the line satisfy the linear equation, indicating a consistent relationship between distance and cost.

Transcripts

how would you determine the cost of a

four mile taxi cab ride

for example

superior cab company charges 80 cents

per mile traveled and a flat rate of two

dollars how much would a four mile cab

ride cost

in this lesson you will learn how to

write and graph a linear function by

examining a real-life scenario with a

linear relationship

let's review a linear relationship is a

relationship between two variables with

a constant rate of change and we should

be able to see that in tables

by looking at the first difference

in graphs as long as we have a straight

line and we'll be able to see that in

real life scenarios as well

let's also remember that slope intercept

form is y equals mx plus b and this

equation of a linear

relationship can be written from a

scenario from a table and from a graph

and to find the slope or that rate of

change the m in this equation

we know we have to go from one point on

the line to the next and to to find the

slope we want to find the change in the

y values and in this case it's negative

three and put that over the change in

the x values which is 6 in this example

and so we end up with a slope of

negative 3 over 6 which is negative

one-half the y-intercept or our starting

point is the b

value in our slope intercept form and we

know that is when the curve or in this

case the line crosses the y axis and

that point is at 0 1 and so when our x

value is 0 whatever our y value is

that's our y intercept and so b equals 1

or our y intercept is 1 here making our

equation in slope-intercept form y

equals negative one-half x plus one

now

looking at a common misunderstanding

standing if we look at an example here

the superior cap company charges 80

cents per mile traveled for a four

mile cab ride you had to pay five

dollars and 20 cents one of the things

that we often make a mistake on is

identifying the information incorrectly

and so yes it's always important to pull

out

the key numbers such as this 80 cents

here and it's clear to us that this is

going to be the constant rate of change

but we could also pick out this four and

this is not the starting point and so

that's not correct and we have to make

sure we understand what information is

given to us through the word problem

because that for and what isn't

highlighted is that five dollars and

twenty cents that's actually going to

give us an ordered pair to the solution

of this linear function

so let's look at a specific example a

taxi cab company charges one dollar and

fifty cents for every mile traveled you

paid a total of seven dollars and twenty

five cents for a cab ride of three and a

half miles well that's great and the

first thing we wanna do is we want to

identify the information provided and

define

what the information is telling us in

this scenario and the very first thing

i'm going to point out here is every

mile traveled that's going to be a clear

indication that we're looking at a

linear scenario because every mile

traveled or per every mile or per mile

is is a really good phrase to identify a

linear relationship because it's telling

us a constant rate of change and so

and 1.50 cents is our constant rate of

change in this scenario

the other key information and we see

that three and a half miles and we see

that you paid

seven dollars and twenty five cents

that's two more pieces of information

that are really important but we need to

make sure we understand what they are

neither of these values are giving us a

starting point what they're giving us is

our independent and dependent variables

to the values of our independent

dependent variables and our independent

variable here is going to be our x-axis

as always but it's the distance traveled

which is three and a half miles and then

the dependent variable which would

always be on our y-axis is our total

cost or 7.25 cents

if we take the information out of that

the constant rate of change being one

and a half

our distance is three and a half and our

cost seven dollars and twenty five cents

i'm going to strip the real world out of

this for a moment and we're going to

just look at the math behind it the

algebra here and that way i know i have

my slope of one and a half and i have my

ordered pair solution to this linear

function

three and a half comma

seven twenty-five hundredths

the three and a half i know is my

independent variable because the

distance traveled therefore it makes it

the x value of my ordered pair the cost

is my dependent variable which makes it

the

y value of my ordered pair and now

i want to model this scenario with an

equation and i already have part of it

and here i define c or my total cost is

equal to one and a half my slope

times d the distance plus something and

that something

is my y-intercept and i'm going to use

the information that i have to find that

y-intercept

so looking at this i have my slope i

have an ordered pair solution to the

linear function and so what i can do as

well my total cost was 7.25 cents the

distance traveled was three and a half

miles i can substitute that into my

partially completed equation and solve

for b so i'm going to do that i'll

substitute seven and a half for c and

three excuse me seven and twenty-five

hundredths for c and three and a half

for

d and i'm gonna go ahead and do the math

and i'll have

7 2500 equals 5 and 25 hundredths plus b

i'm going to subtract 5 and 2500 from

both sides and i'm going to end up with

2 equaling b

and with that information i now have

my entire equation

that will model the linear scenario

about our taxicab ride which is c equals

one half d

plus two

now we can clearly model the scenario

with a

equation but we also can model it with a

table and this will help us identify

some costs for different distances so if

i tr travel

0 miles i can substitute my miles

traveled in for d of my equation and

solve for c and this is basically saying

that once you hop in the cab it's going

to cost you two bucks no matter how far

you travel the next one if i travel one

mile i can substitute that in for d as

well and well the cost of the travel is

one dollar fifty cents but i have to add

that to my flat fee which is two dollars

so i get three dollars and fifty cents

if i travel two miles my total cost will

end up being five dollars because just

traveling is going to cost me three

dollars the one half times two and then

i always have to have my flat rate and

then the last thing i have in my table

here is traveling three miles and if i

travel three miles that's gonna cost me

four dollars and fifty cents but again i

have that flat rate that i have to add

on which gets me to six dollars and 50

cents

now we've modeled this from an equation

our linear scenario with an equation a

table

and we know we can actually graph this

as well and just quickly let's remind

ourselves look at this first difference

every time i increase one mile in my

distance traveled i increase at a

constant rate of one and a half so every

mile i travel i increase my cost of my

ride by a dollar and fifty cents now if

i graph this it's a beautiful graph with

my x-axis being my independent variable

of my distance my y-axis being my

dependent variable of my total cost and

then i have a perfectly straight line

which indicates a linear relationship

but one more step further

there are

more distances i can travel than just

one two three even four five six seven i

can also travel half a mile or two miles

and a quarter you can travel all

distances

and that's what this next part means

these different points on the line are

all going to satisfy this linear

relationship the linear function so all

these points if i were to substitute

them into my linear equation it would

hold true and

these are all different points that

indicate the the number of miles i

travel and what the total cost of that

cab ride will be

in this lesson you have learned how to

write and graph a linear function by

examining a real life scenario with a

linear relationship