The first four terms of a sequence are shown. 16, 48, 144, 432,
what is the common ratio, r, for this sequence?

new question
what is the average rate of change of the function below on the interval from x=-1 and x=1?

g(x)=50(12)x
if necessary, write your answer as a decimal.

new question
which function’s graph has a y-intercept of 1?

question 3 options:

h(x)=0.5(2)x+0.5

h(x)=(0.5)x+1

h(x)=5(2)x

h(x)=5(0.5)x+0.5

new question
which ordered pairs lie on the graph of the exponential function f(x)=2(3)x?

choose all correct answers.

question 1 options:

(2,18)

(0,2)

(−1,1)

(3,56)

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)

∠cbe and ∠deb are same-side interior angles.

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