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

Answer:

(a)Degree 3

(b) $f(x)=x^{3}+2 x^{2}-7 x+3$

Step-by-step explanation:

(a)Given the function represented by the set of points: {(-3,15)(-2,17), (-1,11), (0,3),(1,-1), (2,5),(3,27)}.

To determine the degree of the polynomial of the function, we plot the function on a graph.

From the graph, the function has 2 turning points.

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  where d is the y-intercept.

A cubic function is one in the form $f(x)=a x^{3}+b x^{2}+c x+d$ where d is the y-intercept.

• From the point (0,3) the y-intercept, d=3

Therefore, our polynomial is of the form:

$\begin{array}{l}f(x)=a x^{3}+b x^{2}+c x+3 \\\text {At }(-3,15), 15=a(-3)^{3}+b(-3)^{2}+c(-3)+3 \implies -27 a+9 b-3 c=12 \\\text {At }(-2,17), 17=a(-2)^{3}+b(-2)^{2}+c(-2)+3 \Longrightarrow -8 a+4 b-2 c=14 \\\text {At }(-1,11), 11=a(-1)^{3}+b(-1)^{2}+c(-1)+3 \Longrightarrow -a+b-c=8\end{array}$

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

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

Therefore, the equation for this function is:

$f(x)=x^{3}+2 x^{2}-7 x+3$