We can connect the points you just graphed with a line. To find the equation that represents this linear function, you’ll need the rate of change and the initial amount in the piggy bank. Start by finding the rate of change. In other words, by how much does the number of dollars in the piggy bank change for every week?

Question

t
y
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time in weeks
Dollars in the piggy bank
Rate of change:

Answer 1

To find the rate of change for the linear function represented by the points provided, we can look at how much the dollars in the piggy bank (y) change as time in weeks (t) increases.

From the data you provided, we can see that for every week increase in time (from \(t = 0\) to \(t = 1\), \(t = 1\) to \(t = 2\), etc.), the amount of dollars increases by the same amount.

Let's calculate the rate of change:

1. Identify two points from the table:

   • At \(t = 0\), \(y = 0\)
   • At \(t = 1\), \(y = 3\)

2. Calculate the change in \(y\) and \(t\):

   • Change in \(y\) (Dollars) = \(3 - 0 = 3\)
   • Change in \(t\) (Weeks) = \(1 - 0 = 1\)

3. Calculate the rate of change (slope) using the formula: \[ \text{Rate of change} = \frac{\text{Change in } y}{\text{Change in } t} = \frac{3}{1} = 3 \]

This means the rate of change is 3 dollars per week.

The equation of the linear function can be written in the slope-intercept form \(y = mx + b\), where:

• \(m\) is the slope (rate of change),
• \(b\) is the y-intercept (the initial amount).

From your data:

• The rate of change \(m = 3\)
• The initial amount \(b = 0\) (since when \(t = 0\), \(y = 0\))

Therefore, the equation of the linear function is: \[ y = 3t \]

In conclusion, the rate of change is 3 dollars per week.