To determine the initial value, or the distance Nate started at (usually represented as the y-intercept in a linear function), he can use the information from the table of values he has recorded. Here are a few ways he can find the initial value:
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Check the Time Zero: Since the distance function is linear and has a constant rate of change (50 miles per hour), Nate can look at his table of values for the entry where the time is 0. The distance recorded at that time will be the initial value, representing the distance from the starting point at the very beginning of the trip.
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Use the Equation of the Line: The general form of a linear equation is \( y = mx + b \), where \( y \) is the distance traveled, \( m \) is the rate of change (slope), \( x \) is the time, and \( b \) is the initial value (y-intercept). Here \( m = 50 \) miles per hour. He can plug in a known point from the table (i.e., a specific time \( x \) and the corresponding distance \( y \)). He can then rearrange the equation to solve for \( b \).
For example, if he finds that after 1 hour (x = 1), he is 50 miles away from the starting point (y = 50), then:
\[ 50 = 50(1) + b \]
Simplifying gives:
\[ 50 = 50 + b \implies b = 0 \]
Thus, the initial value is determined to be 0 miles.
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Extrapolate Backward: If Nate has multiple points in his table, he can also extrapolate backward using the rate of change. If he has a recorded distance at time \( t \), he could calculate the distance back to when \( t = 0 \) using the formula:
\[ \text{Distance at } t = 0 = \text{Distance at } t - (50 \times t) \]
This way, for any recorded time and distance, he can find the initial value.
Using any of these methods, Nate will be able to determine the initial value of his distance from the starting point.
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