Answer:
First question: The common ratio r is 3
Second question: The average rate of change is 297.92
Third question: The function's graph of h(x) = 0.5(2^x) + 0.5 has a
y-intercept of 1
Fourth question: The ordered pairs lie on the graph of f(x) are (2 , 18)
and (0 , 2)
Step-by-step explanation:
First question:
* Lets revise the rule of the geometric sequence
- There is a constant ratio between each two consecutive numbers
- Ex:
# 5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)
# 5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)
* General term (nth term) of a Geometric sequence:
- U1 = a  ,  U2  = ar  ,  U3  = ar2  ,  U4 = ar3  ,  U5 = ar4
- Un = ar^n-1, where a is the first term , r is the constant ratio
 between each two consecutive terms  and n is the position
 of the term in the sequence
* Now lets solve the question
∵ The sequence is 16 , 48 , 144 , 432 , ................
∵ a = 16
∵ ar = 48
∴ r = 48/16 = 3
∴ The sequence is geometric withe common ratio 3
* The common ratio r is 3
Second question:
* Lets revise the average rate of change of a function
- When you calculate the average rate of change of a function,
 you are finding the slope of the secant line between the two points
 on the function
- Average Rate of Change  for the function y = f (x) between
 x = a and x = b is:
 change of y/change of x = [f(b) - f(a)]/(b - a)
* Now lets solve the problem
∵ g(x) = 50(12^x), where x ∈ [-1 , 1]
∵ a = -1 and b = 1
∵ f(-1) = 50(12^-1) = 50/12
∵ f(1) = 50(12^1) = 600
∴ Average Rate of Change  = [600 - 50/12]/[1 - (-1)]
∴ Average Rate of Change  = [595.8333]/[2] = 297.92
* The average rate of change is 297.92
Third question:
* Lets talk about the y- intercept
- When any graph intersect the y-axis at point (0 , c), we called
 c the y-intercept
- To find the y- intercept, substitute the value of x in the
 function by zero
* Now lets check which answer will give y- intercept = 1
∵ h(x) = 0.5(2^x) + 0.5 ⇒ put x = 0
∴ h(0) = 0.5(2^0) + 0.5 = 0.5(1) + 0.5 = 1
∵ h(x) = (0.5)^x + 1 ⇒ put x = 0
∴ h(0) = (0.5)^0 + 1 = 1 + 1 = 2
∵ h(x) = 5(2^x) ⇒ put x = 0
∴ h(0) = 5(2^0) = 5(1) = 5
∵ h(x) = 5(0.5)^x + 0.5
∴ h(0) = 5(0.5)^0 + 0.5 = 5(1) + 0.5 = 5.5
* The function's graph of h(x) = 0.5(2^x) + 0.5 has a y-intercept of 1
Fourth question:
* Lets study how to find a point lies on a graph
- When we substitute the value of x of the point in the function
 and give us the same value of y of the point, then the point
 lies on the graph
* Now lets solve the problem
∵ f(x) = 2(3)^x
∵ The point is (2 , 18) ⇒ put x = 2
∴ f(2) = 2(3)² = 2(9) = 18 ⇒ the same y of the point
∴ The point (2 , 18) lies on f(x)
∵ The point is (0 , 2) ⇒ put x = 0
∴ f(0) = 2(3)^0 = 2(1) = 2 ⇒ the same y of the point
∴ The point (0 , 2) lies on f(x)
∵ The point is (-1 , 1) ⇒ put x = -1
∴ f(-1) = 2(3)^-1 = 2(1/3) = 2/3 ⇒ not the same y of the point
∴ The point (-1 , 1) does not lie on f(x)
∵ The point is (3 , 56) ⇒ put x = 3
∴ f(3) = 2(3)³ = 2(27) = 54 ⇒ not the same y of the point
∴ The point (3 , 56) does not lie on f(x)
* The ordered pairs lie on the graph of f(x) are (2 , 18) and (0 , 2)
