x - time(minutes) |8 |5 |7 | 2 |9
y - distance(meters)|40|25|35|10|45
a. every minute ________ meters are traveled
b. write an equation to show the relationship between x and y. _____
c. the relationship is________ because the y-intercept is ________
a. every minute 5 meters are traveled
b. The equation to show the relationship between x and y is y = 5x.
c. The relationship is linear because the y-intercept is 0.
a. To find out how many meters are traveled every minute, we can look at the change in distance (y) for each change in time (x).
Change in distance = Final distance - Initial distance
Change in time = Final time - Initial time
For the first interval:
Change in distance = 25 - 40 = -15
Change in time = 5 - 8 = -3
Since the change in distance is negative and the change in time is negative, we can say that for every minute, -15 meters are traveled.
b. To write an equation to show the relationship between x and y, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.
We can calculate the slope (m) using any two points on the graph:
m = (y2 - y1) / (x2 - x1)
Using the points (8, 40) and (5, 25):
m = (25 - 40) / (5 - 8)
m = -15 / -3
m = 5
Now we know that the equation is y = 5x + b.
To find the value of b (the y-intercept), we can substitute the values of one of the points into the equation and solve for b. Let's use the point (8, 40):
40 = 5(8) + b
40 = 40 + b
b = 0
Therefore, the equation that shows the relationship between x and y is y = 5x.
c. The relationship between x and y is linear because the equation is in the form of a linear equation (y = mx + b), where the variable x has a power of 1. The y-intercept is 0, which means when x is 0 (at the y-intercept), the distance traveled (y) is also 0.
To answer these questions, we need to analyze the given data and look for patterns or trends.
a. To find out how many meters are traveled every minute, we can calculate the difference in distance between consecutive time points and divide it by the difference in time between those points.
Let's take the first two data points: (8, 40) and (5, 25).
The time difference is 8 - 5 = 3 minutes, and the distance difference is 40 - 25 = 15 meters.
So, every minute, (15 meters) / (3 minutes) = 5 meters are traveled.
We can confirm this for the other data points as well:
- For (7, 35) and (5, 25), the time difference is 7 - 5 = 2 minutes, and the distance difference is 35 - 25 = 10 meters. So, every minute, (10 meters) / (2 minutes) = 5 meters are traveled.
- For (2, 10) and (5, 25), the time difference is 2 - 5 = -3 minutes, and the distance difference is 10 - 25 = -15 meters. Note that the time difference is negative because the value of x decreases. So, every minute, (-15 meters) / (-3 minutes) = 5 meters are traveled.
- For (9, 45) and (5, 25), the time difference is 9 - 5 = 4 minutes, and the distance difference is 45 - 25 = 20 meters. So, every minute, (20 meters) / (4 minutes) = 5 meters are traveled.
Therefore, every minute, 5 meters are traveled.
b. To write an equation that shows the relationship between x and y, we can use the formula for the slope of a straight line, which is the change in y divided by the change in x:
slope = (y2 - y1) / (x2 - x1)
Let's choose two data points to calculate the slope, such as (8, 40) and (5, 25):
slope = (40 - 25) / (8 - 5)
slope = 15 / 3
slope = 5
Since the slope is constant, we can write the equation as:
y = mx + b
where m is the slope and b is the y-intercept.
Replacing m with 5 (the slope) gives us:
y = 5x + b
c. The relationship between x and y is linear because the equation we derived, y = 5x + b, is in the form of a linear equation (y = mx + b). The y-intercept (b) is the value of y when x is 0, indicating the starting point. In the given data, when x is 0, y is not provided. Therefore, we cannot determine the y-intercept from the given data.