Question

help please 50 points Tia and Marcus are having a debate about the graph below. Tia says that this is an x^5 function while Marcus claims that it is an x^4 function. Find the fully factored form of this equation and determine who is correct in the debate. You can use the end behavior chart to help you explain your reasoning.

Answer 1

`f(x) = \frac14(x + 1)^2(x + 3)(x -2)( x - 3)`

Tia is correct

Explanation:

From inspection of the graph, we can say that the leading coefficient of the function is positive as:

`f(x) \rightarrow - \infty, \textsf { as } \ x \rightarrow - \infty\\f(x) \rightarrow +\infty, \textsf { as } \ x \rightarrow + \infty`

From inspection of the graph, we can see that the curve intercepts the x-axis at x = -3, x = 2, x = 3

As the curve touches the x-axis at (-1, 0) then this means that the root x = -1 has multiplicity 2

Therefore,
`f(x) = a(x + 1)^2(x + 3)(x -2)( x - 3)` where
`a` is some positive constant.

If we multiply the constants, we can determine the y-intercept:

y-intercept = a x 1² x 3 x -2 x -3 = 18a

Therefore, the y-intercept is at y = 18a

From inspection of the graph, the y-intercept is at y = 4.5

So to determine the value of a: 18a = 4.5 ⇒ a = 1/4

Therefore, the fully factored form of the equation is:

`f(x) = \frac14(x + 1)^2(x + 3)(x -2)( x - 3)`