1) The linear function f(x) contains the points (-10, -29) and (-2, 83).
If g(x) = 25x - 50, which statement is true?
A. The functions f(x) and g(x) both have positive slopes. <<<
B. The functions f(x) and g(x) both have negative slopes.
C. The function f(x) has a positive slope, and the function g(x) has a negative slope.
D. The function f(x) has a negative slope, and the function g(x) has a positive slope.
2) A population of caribou in a forest grows at a rate of 2% every year. If there are currently 237 caribou, which function represents the number of caribou in the forest in t years?
A. C(t) = 237(0.98)^t
B. C(t) = 237(1.02)^t <<<<
C. C(t) = (237)(1.02)^t
D. C(t) = 237(0.02)^t
slope = (83- -29) / (-2 - -10)
= 112/8 = 14
which is indeed positive :)
237 (1.02)^t is correct
1) Using the two given points (-10, -29) and (-2, 83), we can find the slope of the linear function f(x) using the slope formula:
m = (y2 - y1) / (x2 - x1)
For the first point (-10, -29):
x1 = -10
y1 = -29
For the second point (-2, 83):
x2 = -2
y2 = 83
m = (83 - (-29)) / (-2 - (-10))
= (83 + 29) / (-2 + 10)
= 112 / 8
= 14
So the slope of f(x) is 14.
Now let's analyze the function g(x) = 25x - 50.
The slope of g(x) is the coefficient of x, which is 25.
Comparing the slopes:
- The slope of f(x) is 14.
- The slope of g(x) is 25.
Since both slopes are positive, statement A, "The functions f(x) and g(x) both have positive slopes," is true.
Therefore, the correct answer is A.
2) The population of caribou in the forest increases by 2% every year. To represent this growth as an exponential function, we need to multiply the initial population (237) by a factor of (1 + 2%) for each year.
The correct function that represents the number of caribou in the forest in t years is:
C(t) = 237(1.02)^t
Therefore, the correct answer is B.
1) To determine the slope of a linear function, we can use the formula: slope = (change in y)/(change in x).
Given the points (-10, -29) and (-2, 83), we can calculate the change in y and change in x as follows:
Change in y = 83 - (-29) = 112
Change in x = -2 - (-10) = 8
Therefore, the slope of the linear function f(x) is 112/8 = 14.
Now let's analyze the function g(x) = 25x - 50. The coefficient of x, which is 25, is the slope of the function. Therefore, the slope of g(x) is 25.
Comparing the slopes of f(x) and g(x), we can see that 14 < 25. So, the slope of g(x) is greater than the slope of f(x).
Hence, the correct statement is: The functions f(x) and g(x) both have positive slopes. Option A is the answer.
2) The function that represents the number of caribou in the forest in t years can be found using exponential growth. The formula for exponential growth is given as:
A(t) = A_0(1 + r)^t
Where A(t) represents the final amount, A_0 represents the initial amount, r represents the rate of growth, and t represents the time.
In this case, the initial amount is 237 and the rate of growth is 2% or 0.02 (as a decimal). Therefore, the correct expression for the number of caribou in the forest in t years is:
C(t) = 237(1 + 0.02)^t.
Simplifying this expression, we get:
C(t) = 237(1.02)^t.
Option B represents this expression, so the correct answer is B.