Question

The following set of points belong to a specific function: {(-3,15)(-2,17), (-1,11), (0,3),(1,-1), (2,5),(3,27)} Based on the set of points answers the following questions: a) What degree of polynomial function does the set of points represent? Justify your answer. b) Using algebraic methods, write an equation for this function based on the set of points that have been given. Show your work for arriving at the function definition.

Answer 1

(a)Degree 3

(b)
`f(x)=x^3+2x^2-7x+3 .`

Explanation:

The function represented by the set of points: {(-3,15)(-2,17), (-1,11), (0,3),(1,-1), (2,5),(3,27)} has 2 turning points when plotted on a graph.

(a)Now, we know that the maximum number of turning points of a polynomial function is always one less than the degree of the function.

Therefore, the polynomial has a degree of 3

(b)A cubic function is one in the form
`f(x)=ax^3+bx^2+cx+d .` where d is the y-intercept.

From the set of values, the y-intercept, d=3

Therefore, our polynomial is of the form:

`f(x)=ax^3+bx^2+cx+3 .`

`\text{At } (-3,15), 15=a(-3)^3+b(-3)^2+c(-3)+3 \implies -27a+9b-3c=12\\\text{At } (-2,17), 17=a(-2)^3+b(-2)^2+c(-2)+3 \implies -8a+4b-2c=14\\\text{At } (-1,11), 11=a(-1)^3+b(-1)^2+c(-1)+3 \implies -a+b-c=8`

Solving the three resulting equations simultaneously use a calculator), we obtain:

`a=1, b=2, c=-7`

Therefore, an equation of this function is:

`f(x)=x^3+2x^2-7x+3 .`