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Write the trendline equation using function notation

To write the trendline equation using function notation, I'll first analyze the data from the image and then calculate the best-fit line.

### Data extracted:
- Time (sec): 0, 1, 2, 3, 4, 5, 6
- Distance (inches): 0, 6 1/2, 12 7/8, 19 3/8, 25 5/8, 32, 38 5/8

#### Converting distances to decimal:
- \( 6 \frac{1}{2} = 6.5 \)
- \( 12 \frac{7}{8} = 12.875 \)
- \( 19 \frac{3}{8} = 19.375 \)
- \( 25 \frac{5}{8} = 25.625 \)
- \( 38 \frac{5}{8} = 38.625 \)

This gives us the following data points:
- \((0, 0)\)
- \((1, 6.5)\)
- \((2, 12.875)\)
- \((3, 19.375)\)
- \((4, 25.625)\)
- \((5, 32)\)
- \((6, 38.625)\)

#### Next steps:
1. I'll calculate the slope (\(m\)) of the best-fit line using the formula:
   \[
   m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
   \]
2. Then, I'll determine the y-intercept (\(b\)) to finalize the equation.

Let's compute this now.The trendline equation, in function notation, is:

\[
f(t) = 6.415t + 0.040
\]

Here:
- The slope is approximately \(6.415\), indicating that the distance increases by about 6.415 inches for every second.
- The y-intercept is approximately \(0.040\), meaning the initial distance is close to zero (as expected, since at \(t = 0\), \(f(0) \approx 0\)).

Would you like any additional details or clarifications?

### 5 Related Questions:
1. How do you calculate the slope for a set of data points manually?
2. What is the significance of the y-intercept in the context of this trendline?
3. How would the trendline change if the data had larger fluctuations?
4. Can you predict the distance at \(t = 10\) seconds using this equation?
5. What are other methods to determine the accuracy of this trendline?

### Tip:
When interpreting trendlines, always check how well the line fits the data by looking at residuals or calculating the correlation coefficient.

Write the trendline equation using function notation.

Learn how to calculate a linear trendline equation from time and distance data using function notation. The tutorial includes steps to convert fractional distances to decimals, calculate the slope and y-intercept, and finally form the equation f(t) = 6.415t + 0.040. This is suitable for high school students studying linear algebra and statistics.

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