Let's write rules for the functions, g(t) and m(t), first and then we can compare their slopes and y-intercepts. Finally, we will determine their domains and ranges and compare those as well.
Rule for g(t)
A gecko travels at a constant rate of 19 meters per minute. Therefore, in t minutes, it will travel a total distance of
19t meters.
Thus, the function g(t), which represents the distance the gecko travels after t minutes, can be written as:
g(t)=19t.
Rule for m(t)
We must determine the rule for m(t) by looking at the graph. Let's identify the coordinates of two points on the graph to find the slope first!
Now we can substitute (0,0) and (8,300) into the Slope Formula!
m = y_2 - y_1/x_2 - x_1
m = 300 - 0/8 - 0
m = 300/8
m=37.5
The slope of this function is 37.5. Also, because the graph passes through the origin, we know that the y-intercept is 0. Therefore, we can write the final rule for m(t).
m(t)=37.5t
Comparison of slopes and y-intercepts
We now have the rules for the two functions, g(t) and m(t).
g(t)&= 19t
m(t)&= 37.5t
We can see that the slope of m(t) is greater than the slope of g(t). However, both functions have the same y-intercept, y=0.
Domains
A domain is the set of all possible input values for a function. We know that the gecko travels for 6 minutes, this is the maximum time for which it travels. The minimum time is 0 minutes, if it does not travel at all. Then, the domain of g(t) is:
0 ≤ t ≤ 6.
From the graph, we can see that the possible values of t for m(t) start from t=0 and end at t=8. Thus, the domain of m(t) is:
0 ≤ t ≤ 8.
Ranges
Since the domain of g(t) is 0 ≤ t ≤ 6, its range is the set of all real values from g(0) to g(6). Luckily, g(0) is the y-intercept of g(t). From the function rule, and knowing about slope-intercept form, we can tell that it is 0. Let's find g(6) by substituting t=6 in g(t).
g(t)=19t
g( 6)=19( 6)
g(6)=114
The range of g(t) is:
0 ≤ g(t) ≤ 114.
We can identify the range of m(t) by looking at the outputs of m(t) on the graph. The function starts at m(t)=0 and ends at m(t)=300. Therefore, its range is:
0 ≤ m(t) ≤ 300.
