PLLLZZZ HEELLLPPP Ray and Kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. Ray says the third-degree polynomial has four intercepts. Kelsey argues the function can have as many as three zeros only. Is there a way for the both of them to be correct? Explain your answer.

Kelsey has a list of possible functions. Pick one of the g(x) functions below and then describe to Kelsey the key features of g(x), including the end behavior, y-intercept, and zeros.
g(x) = (x + 2)(x − 1)(x − 2)
g(x) = (x + 3)(x + 2)(x − 3)
g(x) = (x + 2)(x − 2)(x − 3)
g(x) = (x + 5)(x + 2)(x − 5)
g(x) = (x + 7)(x + 1)(x − 1)

Create a graph of the polynomial function you selected from Question 2.

Part B

The second part of the new coaster is a parabola.

Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = (x − a)(x − b). Describe the direction of the parabola and determine the y-intercept and zeros.

Create a graph of the polynomial function you created in Question 4.

Part C

Now that the curve pieces are determined, use those pieces as sections of a complete coaster. By hand or by using a drawing program, sketch a design of Ray and Kelsey's coaster that includes the shape of the g(x) and f(x) functions that you chose in the Parts A and B. You do not have to include the coordinate plane. You may arrange the functions in any order you choose, but label each section of the graph with the corresponding function for your instructor to view.

Part A

Please find attached the graph of the function g(x) = (x + 2)·(x - 1)·(x - 2) created with  MS Excel

Part B

Please find attached the parabola created with MS Excel

The shape of the parabola = cup shape

y-intercept is y = 4

The zeros, is x = 2

Part C

Please find attached the graph of the combined coaster created with MS Excel

Step-by-step explanation:

Part A

The maximum number of x-intercepts of a third degree polynomial is between 1 and 3, however, the third degree polynomial can have a y-intercept to make the number of intercepts = 3 + 1 = 4 intercept

a) For the function, g(x) = (x + 2)·(x - 1)·(x - 2), we have;

b) The x-intercepts which are the zeros of the function are;

x = -2, x = 1, and x = 2

The end behavior are;

When x → -∞, y → -∞, x → ∞, y → ∞

The y-intercept is y = (0 + 2)·(0 - 1)·(0 - 2) = 4

The y-intercept is y = 4

(ii) Please find attached the graph of the function created with MS Excel

Part B

The form of the parabola is f(x) = (x - a)·(x - b)

For a y-intercept of 4, we have;

f(x) = (x - 2)·(x - 2) = x² - 4·x + 4

Given that the coefficient of x² is negative, we have;

The parabola is opened up (cup shaped)

The y-intercept = 4

The zero is x = 2

Part C

Please find attached the combined coaster created with MS Excel

hi there!

first we need to find out how much they are adding to each number to make the next number.

ex: 4 + 4 = 8, 8 + 8 = 16, and so on.

so continuing with 32,

32 + 32 = 64

64 + 64 = 128

128 + 128 = 256

256 + 256 = 512

4, 8, 16, 32, 64, 128, 256, 512

the 8th term of the sequence will be 512

hope this ! ~

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752