Question

slope and y - intercepts

• interpret slopes as rates
• interpret y - intercepts as initial values
• interpret slopes as rates

objective 2 - explore proportionality in equations

guided practice activity #2 - proportional equations

this worksheet connects the concepts of proportionality and constant rate to linear functions. please read through the first example that has been done for you, and then complete the following 3 examples.

1. monthly cell phone bill: cost is proportional to minutes you speak at a rate of 5 cents per minute.

a. write an equation for this proportional relationship.
cost = 0.05·minutes
b. now assume there is an additional fixed cost of $20 each month. write down the new equation.
cost₂ = 0.05·minutes + 20

1. weight of water: the weight of water is proportional to the volume at a rate of 8 pounds per gallon.

a. write an equation for this proportional relationship.

b. now assume the bucket you are using to carry the water weighs 10 pounds. write down the new equation.

Explanation:

Step1: Analyze proportional relationship

For a proportional relationship \( y = kx \), where \( k \) is the constant of proportionality. Here, weight \( W \) is proportional to volume \( V \), and \( k = 8 \).
So the equation is \( W=8V \).

Step2: Add fixed - weight (bucket's weight)

When there is a fixed weight of 10 pounds (like the bucket), the new equation is \( W = 8V+10 \) (using the linear equation form \( y=mx + b \), where \( m = 8 \) (slope/rate) and \( b = 10 \) (intercept/initial value)).

Answer:

a. \( W = 8V \)
b. \( W=8V + 10 \)