Question

lesson 4-6
reinforce understanding
linear relationships are modeled by equations. you can compare these
functions by analyzing their equations. each function is represented by the
equation ( y = mx + b ), where ( m ) represents a constant rate of change and ( b
represents an initial value. compare ( m ) and ( b ) to compare the rates of change
and initial values.
ana researches two rental car companies. both charge an administrative
and daily fee. the total cost for downtown rental shop is modeled by
( y = 45x + 50 ). the total cost for uptown rentals is modeled by ( y = 30x + 100 ).
the value of ( m ) is greater for downtown rental shop so that means a greater
rate of change. the value of ( b ) is greater for uptown rentals so that means a
greater initial value. for a given value of ( x ), solve for ( y ) to compare the total costs.
use the description to answer the questions.
an internet provider offers two different plans. both charge an installation fee
and monthly rate. the high-speed plan is modeled by the equation ( y = 63x + 90 ).
the ultimate plan is modeled by ( y = 40x + 130 ).

1. how do the initial values compare?

justify your answer.

1. how do the rates of change compare?

justify your answer.

1. how do the costs of internet for one year compare? justify your answer.
2. a customer is only purchasing internet service for one month. which plan will

be less expensive?

1. the internet provider offers a special rate on the ultimate plan. installation

is now $100. how do the initial values compare now?

1. the internet provider offers a special rate on the high-speed plan. the

monthly rate changes to $59. how do the rates of change compare now?

Explanation:

Question 1

Step1: Identify initial values (b)

In \( y = mx + b \), \( b \) is initial value. For High - Speed: \( b = 90 \), Ultimate: \( b = 130 \).

Step2: Compare b values

Since \( 130>90 \), Ultimate Plan has greater initial value.

Step1: Identify rates of change (m)

In \( y = mx + b \), \( m \) is rate of change. High - Speed: \( m = 63 \), Ultimate: \( m = 40 \).

Step2: Compare m values

Since \( 63>40 \), High - Speed Plan has greater rate of change.

Step1: Determine x for one year

One year has 12 months, so \( x = 12 \).

Step2: Calculate cost for High - Speed Plan

For \( y = 63x+90 \), substitute \( x = 12 \): \( y=63\times12 + 90=756 + 90 = 846 \).

Step3: Calculate cost for Ultimate Plan

For \( y = 40x + 130 \), substitute \( x = 12 \): \( y=40\times12+130 = 480+130 = 610 \).

Step4: Compare costs

Since \( 610<846 \), Ultimate Plan is cheaper for one year.

Answer:

The initial value (installation fee, represented by \( b \) in \( y = mx + b \)) of the Ultimate Plan (\( b = 130 \)) is greater than that of the High - Speed Plan (\( b = 90 \)).

Question 2