To find the equation of the linear function representing the dollars in the piggy bank over time (in weeks), we will first determine the rate of change (slope) and the initial amount (y-intercept).
From your data, the values for \(t\) (time in weeks) and \(y\) (dollars in the piggy bank) are as follows:
t (weeks): 0 1 2 3 4 5 6 7 8 9 10
y (dollars): 0 1.5 3 4.5 6 7.5 9 10.5 12 13.5 15
Step 1: Calculate the Rate of Change
The rate of change (slope) can be found by taking the change in dollars divided by the change in weeks.
Using two points from your data, for example:
- Point 1: (0, 0)
- Point 2: (1, 1.5)
The slope \(m\) can be calculated as follows: \[ m = \frac{\Delta y}{\Delta t} = \frac{1.5 - 0}{1 - 0} = \frac{1.5}{1} = 1.5 \]
You can verify this by checking other intervals, and you’ll find the slope remains consistent.
Step 2: Find the Initial Amount
The initial amount, or the y-intercept (\(b\)), corresponds to the value of \(y\) when \(t = 0\).
From the data, when \(t = 0\):
\[ y = 0 \]
Step 3: Write the Equation
The equation of a linear function in the slope-intercept form \(y = mt + b\) is:
- \(m = 1.5\) (slope)
- \(b = 0\) (y-intercept)
Thus, the equation can be written as:
\[ y = 1.5t + 0 \] or simply: \[ y = 1.5t \]
Conclusion
The equation that represents the linear function for the amount of dollars in the piggy bank over time in weeks is:
\[ y = 1.5t \]
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