Someone please help me with this!!!
Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the coaster's track.
Part A
The first part of Ray and Kelsey's roller coaster is a curved pattern that can be represented by a polynomial function.
Ray and Kelsey are working to graph a third-degree polynomial function that represents the first pattern in the coaster plan. Ray says the third-degree polynomial has 4 intercepts. Kelsey argues the function can have as many as 3 zeros only. Is there a way for the both of them to be correct? Explain your answer.
Kelsey has a list of possible functions. Pick one of the g(x) functions below and then describe to Kelsey the key features of g(x), including the end behavior, y-intercept, and zeros.
g(x) = x3 − x2 − 4x + 4
g(x) = x3 + 2x2 − 9x − 18
g(x) = x3 − 3x2 − 4x + 12
g(x) = x3 + 2x2 − 25x − 50
g(x) = 2x3 + 14x2 − 2x − 14
Ray and Kelsey are discussing the number of intercepts and zeros in a third-degree polynomial function that represents the first pattern in the roller coaster's plan.
Ray says that the third-degree polynomial has 4 intercepts. However, Kelsey argues that the function can have as many as 3 zeros only.
In this case, both Ray and Kelsey can be correct. This is because intercepts and zeros are related but not exactly the same.
Intercepts are the points where the graph of a function intersects the x-axis. These points represent the values of x where the function's output, y, is equal to zero. So, if the third-degree polynomial function has 4 intercepts, it means that there are 4 different values of x where the function equals zero.
On the other hand, zeros are the values of x where the function equals zero. The degree of a polynomial function determines the maximum number of zeros it can have. A polynomial function of degree n can have at most n zeros. Since a third-degree polynomial is of degree 3, it can have at most 3 zeros.
Now, let's pick one of the g(x) functions and describe its key features:
Let's pick g(x) = x^3 - x^2 - 4x + 4.
End behavior: As x approaches positive infinity or negative infinity, the value of g(x) also approaches positive infinity. This is because the leading term x^3 dominates the function.
Y-intercept: To find the y-intercept, we set x = 0 and evaluate g(x). So, g(0) = 0^3 - 0^2 - 4(0) + 4 = 4. Therefore, the y-intercept is (0, 4).
Zeros: To find the zeros of the function, we set g(x) = 0 and solve for x. By factoring or using another method, we find that the zeros of g(x) = x^3 - x^2 - 4x + 4 are x = -1 and x = 2.
Therefore, the key features of g(x) = x^3 - x^2 - 4x + 4 are: end behavior approaches positive infinity, y-intercept is (0, 4), and zeros are x = -1 and x = 2.
To answer if Ray and Kelsey can both be correct, let's first clarify a few terms.
A zero of a polynomial function is a value for which the function evaluates to zero. In other words, it's a value that makes the polynomial equation true. Zeros are also known as roots or x-intercepts.
The degree of a polynomial function is the highest power of the variable in the function. For example, a third-degree polynomial has the highest power of x as 3.
Now, consider Ray's claim that a third-degree polynomial has 4 intercepts. This statement is incorrect. The number of intercepts or zeros of a polynomial function is determined by its degree. A third-degree polynomial can have up to 3 zeros, but not more.
On the other hand, Kelsey's claim that a third-degree polynomial can have as many as 3 zeros is correct. The fundamental theorem of algebra states that a polynomial function of degree n can have at most n zeros. Therefore, a third-degree polynomial can have up to 3 zeros.
Now let's analyze one of the given functions, g(x) = x^3 − x^2 − 4x + 4, to describe its key features.
End behavior: As the leading term of the polynomial is x^3, which is an odd-degree term with a positive coefficient, the end behavior of the function will be:
- As x approaches negative infinity, g(x) will also approach negative infinity.
- As x approaches positive infinity, g(x) will also approach positive infinity.
Y-intercept: The y-intercept is the value of the function when x is 0. Plugging in x = 0 into g(x) = x^3 − x^2 − 4x + 4, we get g(0) = 4. Therefore, the y-intercept is 4.
Zeros: Zeros or x-intercepts are the values of x for which g(x) = 0. To find the values of x that make g(x) equal to zero, we can use various methods like factoring, synthetic division, or numerical methods. In this case, the zeros can be found by factoring g(x). By factoring, we get:
g(x) = (x-1)(x+2)(x-2)
This means that the function has three zeros: x = 1, x = -2, and x = 2.
So, for the given function g(x) = x^3 − x^2 − 4x + 4:
- The end behavior is both positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
- The y-intercept is 4.
- The zeros are x = 1, x = -2, and x = 2.
I hope this explanation helps you understand the concepts and answer the questions. If you have any further doubts, feel free to ask!
the y-intercept is the constant term
the end behavior is y->∞ or y-> -∞, as with all polynomials.
the only other features of interest are the x-intercepts. A little synthetic division will reveal those.