help me please with question i have needed help for hours

Question

Question 1:The number of cars sold per month can be modeled by f(x) = 0.03x³ + 0.4x² + 0.7x + 75, where x represents the number of months since April. Let g(x) = f(x) − 25. Which of the following gives the rule for g and the correct explanation of the meaning of the transformation in terms of monthly car sales?

A.) 0.03x3 + 0.4x2 + 0.7x + 50

vertical shift 25 units down

decrease of 25 units per month

B.) −0.03x3 + 0.4x2 − 0.7x + 100

vertical shift 25 units up

increase of 25 units per month

C.) 0.03x³ + 0.4x² + 0.7x + 100

vertical shift 25 units up

increase of 25 units per month

D.) 0.03x3 + 0.4x2 + 0.7x + 25

vertical shift 50 units down

decrease of 50 units per month

Question 2: The table shows the numbers of cars manufactured by a company for selected years. Identify a polynomial function for cars manufactured in thousands where x represents the years since 1990.

A.) P(x) = 0.2x³ + 0.5x² + 0.4x + 0.3

B.) P(x) = 0.5x³ + 0.5x² + 0.4x + 0.4

C.) P(x) = 0.2x³ + 0.2x² + 0.3x + 0.3

D.) P(x) = 0.5x³ + 0.2x² + 0.3x + 0.4

Question 3: Use finite differences to identify the degree of the polynomial that best describes the data.

A.) quartic

B.) cubic

C.) quadratic

D.) linear

Please just give and only give the explanation and the solution.

Answer 1

Question 1: g(x) = f(x) -25 = (0.03x³ + 0.4x² + 0.7x + 75) - 25 = 0.03x³ + 0.4x² + 0.7x + 50.

So the answer ia A).

Question 2: Since x represents the years since 1990, at the years 1991, 1993, 1995, 1995, 1997, 1999, x = 1, 3, 5, 7, 9. Therefore, P(1) = 1.4, P(3) = 11.4, P(5) = 39.8, P(7) = 96.2 and P(9) = 190.2.

Check this condition, only P(x) = 0.2x³ + 0.5x² + 0.4x + 0.3 satisfies. So the answer is A).

Question 3:

x	-3	-1	1	3	5	7
y	4	6	1	-11	-30	-56

First Differences. 6 - 4 = 2 1 - 6 = -5 -11 - 1 = -12 -30 - (-11) = -19 -56 - (-30) = -26

Second Differences. -5 - 2 = -7 -12 - (-5) = -7 -19 - (-12) = -7 -26 - (-19) = -7

A second degree polynomial best describes the data. So the answer is C)