Investigation 8.1 Interpolating Polynomials
In Chapter 6, we learned to fit a quadratic function through three points on its graph. A polynomial whose graph passes through a given set of points is called an interpolating polynomial. Because polynomials are easy to evaluate and manipulate, they are often used to approximate more complicated functions and to describe the shapes of curves.
In this Investigation, we find interpolating polynomials of degrees \(1\text{,}\) \(2\text{,}\) and \(3\) to approximate the function \(f(x) = \dfrac{12}{x}\text{.}\)

Graph the function \(f(x) = \dfrac{12}{x}\) in the window
\begin{align*} \text{Xmin} \amp = 2 \amp\amp \text{Xmax} = 7.4\\ \text{Ymin} \amp = 5 \amp\amp \text{Ymax} = 20 \end{align*} 
Complete the table of values.
\(x\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(f(x)\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)


First we will find a linear polynomial \(P_1(x) = ax + b\) that matches \(f(x)\) at \(x = 1\) and \(x=6\text{.}\) We must find constants \(a\) and \(b\) so that \(P_1(1) = f(1)\) and \(P_1(6) = f(6)\text{.}\)

The two conditions above translate into equations about \(a\) and \(b\text{.}\) The constants \(a\) and \(b\) must satisfy the system
\begin{align*} a \cdot 1 + b \amp = f (1)\\ a \cdot 6 + b \amp = f (6) \end{align*}Solve the system and find the polynomial \(P_1(x)\text{.}\)
Graph \(P_1(x)\) in the same window with \(f(x)\text{.}\)


Next, we will find a quadratic polynomial \(P_2(x) = ax^2 + bx + c\) that matches \(f(x)\) at \(x = 1\text{,}\) \(2.5\text{,}\) and \(6\text{.}\)
Write and solve a system of equations for the constants \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\)
Graph \(P_2(x)\) in the same window with \(f(x)\text{.}\)

We can also find a cubic polynomial \(P_3(x) = ax^3 + bx^2 + cx + d\) that matches \(f(x)\) at \(x = 1\text{,}\) \(3\text{,}\) \(4\text{,}\) and \(6\text{.}\)
Write a system of equations for \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\) We will see how to solve such a system later in this chapter. For now, we will use the calculator's cubic regression feature.
Enter the coordinates of \(P_3(x)\) evaluated at \(x=1\text{,}\) \(3\text{,}\) \(4\text{,}\) and \(6\) into \(L_1\) and \(L_2\) under the STAT EDIT menu. Then, from the STAT CALC menu, choose 6: CubicReg and press
ENTER
.Graph \(P_3(x)\) in the same window with \(f(x)\text{.}\)

How well does each interpolating polynomial approximate the function \(f(x)\text{?}\) Graph the "error function," \(E_n(x)\text{,}\) for each polynomial on the interval \([2, 7.4]\text{.}\)
\begin{align*} E_1(x) \amp = f(x)  P_1(x)\\ E_2(x) \amp = f(x)  P_2(x)\\ E_3(x) \amp = f(x)  P_3(x) \end{align*}(You will have to choose a suitable \(y\)window for each error function.) What is the maximum error on the interval \([2, 7.4]\) for each approximating polynomial?