4. Given the table for x, time in minutes, and y, distance traveled!
x: time
8
5
7
2
9
y: distance
664
415
581
166
747
(4 points)
a Every minuets____ meters are traveled
b. Write an equation to show the relationship between x and y is ____
c. This relationship is _____ because the y-intercept is
a. To find the rate at which meters are traveled every minute, we can calculate the difference in distance (y) for each consecutive pair of time values (x).
For the given values, the differences in distance are:
- Between x = 8 and x = 5: 664 - 415 = 249 meters
- Between x = 5 and x = 7: 415 - 581 = -166 meters
- Between x = 7 and x = 2: 581 - 166 = 415 meters
- Between x = 2 and x = 9: 166 - 747 = -581 meters
Therefore, the distances traveled for each minute of time vary and are not constant.
b. To write an equation to show the relationship between x and y, we can examine the pattern and determine the best fit function. However, without additional information or context, it is not possible to determine the specific relationship between the time (x) and distance (y) values.
c. Without knowing the equation or context, we cannot determine if the relationship is linear or non-linear. Similarly, we cannot determine the y-intercept without additional information or context.
a) To find the distance traveled every minute, we can calculate the difference in distance between consecutive time points.
For example, between time 8 and time 5, the distance traveled is 664 - 415 = 249 meters.
Similarly, between time 5 and time 7, the distance traveled is 415 - 581 = -166 meters.
Between time 7 and time 2, the distance traveled is 581 - 166 = 415 meters.
Lastly, between time 2 and time 9, the distance traveled is 166 - 747 = -581 meters.
So, we have the following distances traveled every minute: 249 meters, -166 meters, 415 meters, -581 meters.
b) To write an equation to show the relationship between x (time) and y (distance), we need to consider the pattern or trend in the data. Looking at the distances traveled every minute, we can observe that the pattern alternates between positive and negative differences.
One possible equation to represent this relationship is y = mx + b, where m represents the average change in y (distance) for every one unit change in x (time), and b represents the y-intercept.
Since the pattern alternates between positive and negative differences, we can consider two separate equations depending on the value of x.
For x <= 5:
y = mx + b1
For x >= 7:
y = mx + b2
c) The relationship can be classified as linear if the y-intercept is a constant value for all values of x. To determine whether this is the case, we need to find the y-intercepts for both equations.
To find the y-intercept, we can substitute x = 0 into each equation.
For x <= 5:
y = m(0) + b1
=> y = b1
For x >= 7:
y = m(0) + b2
=> y = b2
Since both equations have the same form for y-intercept (b1 and b2), it indicates that the y-intercept is the same regardless of the value of x. Therefore, the relationship is linear.
To find the answers to the questions:
a) To calculate the distance traveled per minute, we need to find the difference between the distances corresponding to consecutive time values and divide it by the difference in time. Let's take the first and second time-distance pairs: (8, 664) and (5, 415).
Distance traveled in 3 minutes = 664 - 415 = 249 meters
So, every minute 249/3 = 83 meters are traveled.
b) To write an equation representing the relationship between x (time) and y (distance), we need to determine the mathematical form that relates the two variables. Let's try to find a linear equation in the form y = mx + b, where m represents the slope and b represents the y-intercept.
By observation, we can see that if we take the first and last time-distance pairs: (8, 664) and (9, 747), the change in distance is 747 - 664 = 83 meters, which coincides with the value we obtained in part a).
Since the slope (m) represents the change in y over the change in x, we have:
m = (747 - 664) / (9 - 8) = 83
To determine the y-intercept (b), we can choose any point on the line, let's take (8, 664), and substitute its values into the linear equation:
y = mx + b
664 = 83(8) + b
664 = 664 + b
b = 0
So, the equation that represents the relationship between x and y is y = 83x + 0, which simplifies to y = 83x.
c) The relationship is linear because the y-intercept (b) is 0. In a linear equation, if the y-intercept is 0, it means that the line passes through the origin (0,0) on the coordinate plane.