Which of the formulas below could be a polynomial with all of the following properties: its only zeros are x = -6, -2, 2, it has y-intercept y = 4, and its long-run behavior is y rightarrow - infinity as x plusminus infinity? select every formula that has all of these properties. a. y = -4/144 (x + 6)^2 (x + 2)(x - 2) b. y = -4/192 (x + 6)(x + 2)^4 (x - 2) c. y = -4x (x + 6)(x + 2)(x - 2) d. y = -4/24 (x + 6)(x + 2)(x - 2) e. y = -4/48 (x + 6) (x + 2)^2 (x - 2) f. y = -4/48 (x + 6)(x + 2)(x - 2)^2 g. y = 4/48 (x + 6)(x + 2) (x - 2)^2

A, B, and E

if I read your functions right.

Step-by-step explanation:

It's zeros are x=-6,-2, and 2.

This means we want the factors (x+6) and (x+2) and (x-2) in the numerator.

It has a y-intercept of 4.  This means we want to get 4 when we plug in 0 for x.

And it's long-run behavior is y approaches - infinity as x approaches either infinity.  This means the degree will be even and the coefficient of the leading term needs to be negative.

So let's see which functions qualify:

A) The degree is 4 because when you do x^2*x*x you get x^4.

The leading coefficient is -4/144 which is negative.

We do have the factors (x+6), (x+2), and (x-2).

What do we get when plug in 0 for x:

Put into calculator:  4

A works!

B) The degree is 6 because when you do x*x^4*x=x^6.

The leading coefficient is -4/192 which is negative.

We do have factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

Put into calculator: 4

B works!

C) The degree is 4 because when you do x*x*x*x=x^4.

The leading coefficient is -4 which is negative.

Oops! It has a zero at 0 because of that factor of (x) between -4 and (x+6).

So C doesn't work.

D) The degree is 3 because x*x*x=x^3.

We needed an even degree.

D doesn't work.

E) The degree is 4 because x*x^2*x=x^4.

The leading coefficient is -4/48 which is negative.

It does have the factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

Put into calculator: 4

So E does work.

F) The degree is 4 because x*x*x^2=x^4.

The leading coefficient is -4/48.

It does have factors (x+6), (x+2), and (x-2).

What do we get when we plug in 0 for x:

Put into calculator: -4

So F doesn't work.

G. I'm not going to go any further. The leading coefficient is 4/48 and that is not negative.

So G doesn't work.

Step-by-step explanation:

Notice that we have 3 zeros, which means there are only 3 roots, which are -6, -2 and 2, this indicates that our expression must be cubic with the binomials (x+6), (x+2) and (x-2).

We this analysis, possible choices are C and D.

Now, according to the problem, it has y-intercept at y = 4, so let's evaluate each expression for x = 0.

Therefore, choice D is the right expression because it has all given characteristics.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

a

step-by-step explanation:

the answer is a