Write and graph a linear function by examining a real-life scenario
Summary
TLDRThis 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
- π The cost of a taxi ride can be calculated using a linear function with a fixed rate and a flat fee.
- π£οΈ Superior Cab Company charges $0.80 per mile with a $2 flat rate, making a 4-mile ride cost $5.20.
- π A linear relationship has a constant rate of change and can be represented using a table, graph, or equation.
- π’ 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).
- π 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.
- π Real-life scenarios, like taxi fares, demonstrate linear relationships where cost changes consistently with distance traveled.
- πΈ For example, a taxi company charging $1.50 per mile plus a flat fee can be modeled with an equation.
- π Using known values like distance traveled and cost, you can substitute them into the equation to solve for missing variables, like the y-intercept.
- π Tables can be used to calculate the total cost for different distances by plugging values into the linear equation.
- π 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
π 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.
π’ 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.
Mindmap
Keywords
π‘Linear Function
π‘Slope-Intercept Form
π‘Rate of Change
π‘Y-Intercept
π‘Independent Variable
π‘Dependent Variable
π‘Constant Rate
π‘Ordered Pair
π‘Table of Values
π‘Graph
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
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