Question

The shape of a rollercoaster is modeled by a polynomial function, R(x). Describe how to find the x-intercepts of R(x) and how to construct a rough graph of R(x) so that the engineer can predict when there will be no change in the direction of the coaster. You may create a sample polynomial to be used in your explanations.

Answer 1

The first feature of a polynomial function is that its graph is continuous, that is, the graph of a polynomial function has no breaks. The second feature is that the graph of a polynomial function has only smooth, rounded turns.

Since the problem establishes that we may create a sample polynomial to be used in our explanations, we can say that this polynomial function is:

`R(x)=-2x^4+2x^2`

To find the x-intercepts of R(x) set R(x) equal to zero and solve for x, so:

`Set \ R(x) \ equal \ to \ zero: \\ \\ -2x^4+2x^2=0 \\ \\ Remove \ common \ monomial \ factor: \\ \\ -2x^2(x^2-1)=0 \\ \\ Factor \ out: \\ \\ -2x^2(x-1)(x+1)=0`

So the real zeros are:

`\boxed{x=0} \\ \\ \boxed{x=1} \\ \\ \boxed{x=-1}`

To construct a rough graph of R(x) we start setting the zeros on the x-axis. On the other hand, given that the leading coefficient
`a_(n)=-2` is negative, that is,
`a_(n)<0`. The graph falls to the left and right. So the graph is shown in the Figure below.

Answer 2

You will need three roots for this, so we have

Let x = -30, -10 and +20

So the factors will be (x+30)(x+10)(x-20)

The divide it to 100, this will help bring the peak up and down

So the polynomial function R(x) will become

1/100 * (x+30)(x+10)(x-20)

R(x) = 1/100 * (x+30)(x+10)(x-20)

Finding the X-intercept:

Let R(x) = 0 and solve for x.

1/100 * (x+30)(x+10)(x-20) = 0

x = -30, -10, 20 are the x-intercepts.