1. The function f(x) passes through the points (−4, 6) and (−2, 3)
The function g(x) is represented by the equation g(x)=1/2x^2−3
Which of the following statements is true of these two functions?
A. On the interval [−4,−2] the average rate of change for f(x) is equal to −2 which means that f(x) is changing at a faster rate than g(x)
B. On the interval [−4,−2] the average rate of change for f(x) is equal to −3/2 which means that f(x) is changing at a faster rate than g(x)
C. On the interval [−4,−2] the average rate of change for g(x) is equal to 3 which means that g(x) is changing at a faster rate than f(x)
D. On the interval [−4,−2] the average rate of change for g(x) is equal to −3 which means that g(x) is changing at a faster rate than f(x)
First, let's find the average rate of change for f(x) on the interval [-4, -2]:
To find the average rate of change, we need to calculate the difference in the y-values divided by the difference in the x-values.
For f(x), the given points are (-4, 6) and (-2, 3). The difference in the y-values is 3 - 6 = -3. The difference in the x-values is -2 - (-4) = 2.
So, the average rate of change for f(x) on the interval [-4, -2] is -3/2.
Now let's calculate the average rate of change for g(x) on the same interval:
To find the average rate of change for g(x), we need to calculate the difference in the y-values divided by the difference in the x-values.
For g(x), the given equation is g(x) = 1/2x^2 - 3.
Plugging in the x-values -4 and -2 into the equation, we get g(-4) = 1/2(-4)^2 - 3 = 8 - 3 = 5, and g(-2) = 1/2(-2)^2 - 3 = 2 - 3 = -1.
The difference in the y-values is -1 - 5 = -6. The difference in the x-values is -2 - (-4) = 2.
So, the average rate of change for g(x) on the interval [-4, -2] is -6/2 = -3.
Now let's compare the average rate of change for f(x) and g(x) on the interval [-4, -2]:
From the calculations above, we can see that the average rate of change for f(x) on the interval [-4, -2] is -3/2, and the average rate of change for g(x) on the same interval is -3.
Comparing the two, we can conclude that the following statement is true:
D. On the interval [−4,−2] the average rate of change for g(x) is equal to −3 which means that g(x) is changing at a faster rate than f(x)