Question

name functions equations practice
learning target success criteria
cc.8.fa.5 - construct a function to model a linear relationship between two quantities. determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
cc.8.fa.3 - interpret the equation y = mx + b as defining a linear function, whose graph is a straight line
i can determine the rate of change and the initial value of a function from a graph and/or table
i can interpret the rate of change and the initial value of a function from a graph and/or table.
i can interpret the equation of a linear function from a table or graph.
directions: read the following questions and find the rate of change, initial value and then write the equation in y=mx+b form.

1. bowling alley

a bowling alley charges a shoe rental fee and an hourly lane fee.
the total cost for 1 hour is $14 and for 4 hours is $32.
assume the relationship is linear.
rate of change
m = (32 - 14)/(4 - 1) = 18/3 = 6 per hour

1. bike rental shop

a bike rental shop charges a flat rental fee plus an hourly rate.
the total cost for 3 hours is $22 and for 6 hours is $40.
assume the relationship is linear.
rate of change
m = (40 - 22)/(6 - 3) = 18/3

1. tutoring service

a tutoring service charges an initial sign - up fee and an hourly tutoring rate.
the total cost for 2 hours is $55 and for 5 hours is $100.
assume the relationship is linear.
rate of change
m = (100 - 55)/(5 - 2) = 45/3 = 15 per hour

Explanation:

Problem 2: Bike Rental Shop

Step 1: Find the rate of change (slope \( m \))

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(3, 22)\) and \((x_2, y_2)=(6, 40)\).
\[
m=\frac{40 - 22}{6 - 3}=\frac{18}{3}=6
\]

Step 2: Find the initial value (y - intercept \( b \))

Use the linear equation \( y = mx + b \) and substitute \( m = 6 \), \( x = 3 \), \( y = 22 \).
\[
22=6\times3 + b
\]
\[
22 = 18 + b
\]
Subtract 18 from both sides:
\[
b=22 - 18 = 4
\]

Step 3: Write the equation in \( y=mx + b \) form

Substitute \( m = 6 \) and \( b = 4 \) into \( y=mx + b \).
\[
y = 6x + 4
\]

Step 1: Find the rate of change (slope \( m \))

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(2, 55)\) and \((x_2, y_2)=(5, 100)\).
\[
m=\frac{100 - 55}{5 - 2}=\frac{45}{3}=15
\]

Step 2: Find the initial value (y - intercept \( b \))

Use the linear equation \( y = mx + b \) and substitute \( m = 15 \), \( x = 2 \), \( y = 55 \).
\[
55=15\times2 + b
\]
\[
55 = 30 + b
\]
Subtract 30 from both sides:
\[
b=55 - 30 = 25
\]

Step 3: Write the equation in \( y=mx + b \) form

Substitute \( m = 15 \) and \( b = 25 \) into \( y=mx + b \).
\[
y = 15x + 25
\]

Answer:

• Rate of change: \( 6 \) dollars per hour
• Initial value: \( 4 \) dollars
• Equation: \( y = 6x + 4 \)

Problem 3: Tutoring Service