Question
Compare the linear function and the arithmetic sequence below Linear Function xy 1 10 0 14 1 18 2 22 The rate of change of the linear function is The y intercept of the linear function is Arithmetic Sequence Explicit formula an 10 4 n 1 an 10 4 n 1 The rate of change common difference of the arithmetic sequence is The first value of the arithmetic sequence is The rate of change of this linear function is the rate of change of this arithmetic sequence Arithmetic sequences can be represented as a linear function
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GPT-4o mini
To compare the linear function and the arithmetic sequence, we can break down the information you've provided.
### Linear Function
From the linear function you provided:
- Points: (0, 14), (1, 18), (2, 22), and (1, 10) seems contradictory to the other pairs.
We can interpret this data. Let’s calculate the slope (rate of change) from the two consistent points:
Using the points (0, 14) and (1, 18):
- Slope (m) = (y2 - y1) / (x2 - x1) = (18 - 14) / (1 - 0) = 4.
Using the points (1, 18) and (2, 22):
- Slope (m) = (22 - 18) / (2 - 1) = 4.
Thus, the **rate of change of the linear function** is **4**.
### Y-Intercept
The **y-intercept** is the y-value when x = 0. From the point (0, 14), we find that the **y-intercept of the linear function** is **14**.
### Arithmetic Sequence
The explicit formula for the arithmetic sequence is given as:
\[ a_n = 10 + 4(n - 1) \]
#### Rate of Change
- The **common difference** (rate of change) is **4**, as indicated by the "+ 4" term in the formula.
#### First Value
To find the **first value** of the arithmetic sequence, we can simply set \( n = 1 \):
\[ a_1 = 10 + 4(1 - 1) = 10 + 4 \cdot 0 = 10 \]
So, the **first value of the arithmetic sequence** is **10**.
### Summary
- **Rate of Change of Linear Function**: **4**
- **Y-Intercept of Linear Function**: **14**
- **Rate of Change of Arithmetic Sequence**: **4**
- **First Value of Arithmetic Sequence**: **10**
### Conclusion
Both the linear function and the arithmetic sequence have the same rate of change of **4**. The arithmetic sequence can be represented as a linear function, specifically in the format of \( y = mx + b \) where \( m \) is the rate of change and \( b \) is the y-intercept.
### Linear Function
From the linear function you provided:
- Points: (0, 14), (1, 18), (2, 22), and (1, 10) seems contradictory to the other pairs.
We can interpret this data. Let’s calculate the slope (rate of change) from the two consistent points:
Using the points (0, 14) and (1, 18):
- Slope (m) = (y2 - y1) / (x2 - x1) = (18 - 14) / (1 - 0) = 4.
Using the points (1, 18) and (2, 22):
- Slope (m) = (22 - 18) / (2 - 1) = 4.
Thus, the **rate of change of the linear function** is **4**.
### Y-Intercept
The **y-intercept** is the y-value when x = 0. From the point (0, 14), we find that the **y-intercept of the linear function** is **14**.
### Arithmetic Sequence
The explicit formula for the arithmetic sequence is given as:
\[ a_n = 10 + 4(n - 1) \]
#### Rate of Change
- The **common difference** (rate of change) is **4**, as indicated by the "+ 4" term in the formula.
#### First Value
To find the **first value** of the arithmetic sequence, we can simply set \( n = 1 \):
\[ a_1 = 10 + 4(1 - 1) = 10 + 4 \cdot 0 = 10 \]
So, the **first value of the arithmetic sequence** is **10**.
### Summary
- **Rate of Change of Linear Function**: **4**
- **Y-Intercept of Linear Function**: **14**
- **Rate of Change of Arithmetic Sequence**: **4**
- **First Value of Arithmetic Sequence**: **10**
### Conclusion
Both the linear function and the arithmetic sequence have the same rate of change of **4**. The arithmetic sequence can be represented as a linear function, specifically in the format of \( y = mx + b \) where \( m \) is the rate of change and \( b \) is the y-intercept.
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