Question

Analyze the polynomial function(b) The x-intercept(s) is/are(c) The y-intercept is(d) Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept.The zero(s) of f is/are(f) Determine the maximum number of turning points on the graph of the function.nothing

Answer 1

(a) If we expand the polynomial we would get a cubic function with 1 as the leading coefficient, then the end behavior of the graph is like

`y=x^3`

(b) At the x-intercepts the value of f(x) is zero, that is,

`0=(x+4)(x-5)(x+6)`

then, the terms inside the parenthesis must be zero. This is obtained with the next values:

x = -4

x = 5

x = -6

The points are: (-4, 0) (5, 0) and (-6, 0)

(c) At the y-intercept, x = 0. Substituting it into the formula we get:

`\begin{gathered} f(x)=(x+4)(x-5)(x+6) \\ f(0)=(0+4)(0-5)(0+6) \\ f(0)=4\cdot(-5)\cdot6 \\ f(0)=-120 \end{gathered}`

The point is (0, -120)

(d) The zeros of the function are the x-intercepts already mentioned in item b. All of them are multiplicity one because all the terms in parenthesis are raised to the power of 1. Since all of them are multiplicity 1, then all of them crosses the x-axis

(e) The least zero is a zero of multiplicity 1 , so the graph of f crosses the x-axis at x = -6.

The middle zero is a zero of multiplicity 1 , so the graph of f crosses the x-axis at x = -4.

The greatest zero is a zero of multiplicity 1 , so the graph of f crosses the x-axis at x = 5.

(f) Given that the function behaves like a cubic function and there are 3 zeros, then the number of turning points is 2

(g) Choices A and B behaves like

`y=-x^3`

then, they are discarded.

The difference between choices C and D is the y-intercept. We can see that in choice D, the graph intersects the y-axis near y = -100, and in choice C, the intersection is a greater value, closer to 0. In conclusion, choice D is correct