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Section 6.6 Curve Fitting

Subsection 1. Write an equation for a point on a graph

If a curve passes through a given point, the coordinates of the point satisfy the equation of the curve.

Subsubsection Example

Example 6.86.

Write an equation to say that \((-3,8)\) lies on the graph of\(~y=ax^2+bx+c~\text{.}\)

Solution

Substitute \(-3\) for \(x\) and \(8\) for \(y\text{.}\)

\begin{align*} 8 \amp = a(-3)^2+b(-3)+c \amp \amp \blert{\text{Simplify.}}\\ 8 \amp = 9a-3b+c \end{align*}

Subsubsection Exercises

Write an equation to say that \((-4,-18)\) lies on the graph of\(~y=ax^2+bx+c~\text{.}\)

Answer
\(-16a-4b+c=-18\)

Write an equation to say that \((8,0)\) lies on the graph of\(~y=ax^2+bx+c~\text{.}\)

Answer
\(64a+8b+c=0\)

Write an equation to say that \((0,-5)\) lies on the graph of\(~y=ax^2+bx+c~\text{.}\)

Answer
\(c=-5\)

Write an equation to say that \((-60,400)\) lies on the graph of\(~y=ax^2+bx+c~\text{.}\)

Answer
\(3600a-60b+c=400\)

Subsection 2. Solve a 2x2 linear system

For fitting a parabola through given points, we'll solve systems using the method of elimination.

Subsubsection Example

Example 6.91.

Solve the system by elimination.

\begin{align*} 5x-2y \amp = 22\\ 2x-5y \amp = 13 \end{align*}
Solution

To eliminate the \(x\)-terms,look for the smallest integer that both 2 and 5 divide into evenly, namely, 10. Multiply the first equation by 2 and the second equation by \(-5\text{.}\)

\begin{align*} \blert{2}(5x-2y \amp = 22) \amp\amp \rightarrow \amp 10x-4y \amp = 44\\ \blert{-5}(2x-5y \amp = 13) \amp\amp \rightarrow \amp -10x+25y \amp = -65 \end{align*}

Add these new equations to obtain an equation in \(y\text{.}\)

\begin{align*} 10x-~4y \amp = 44\\ \underline{-10x+25y} \amp \underline{{}= -65 \vphantom{-10x+25y}}\\ 21y \amp = -21 \end{align*}

Solve for \(y\) to find \(y=-1\text{.}\) Finally, substitute \(y=\alert{-1}\) into the first equation and solve for \(x\text{.}\)

\begin{align*} 5x-2(\alert{-1}) \amp = 22\\ 5x+2 \amp = 22\\ x \amp = 4 \end{align*}

The solution to the system is \((4,-1)\text{.}\)

Subsubsection Exercises

Solve the system by elimination.

\begin{align*} 2x-9y \amp = 3\\ 4x-5y \amp = -7 \end{align*}
Answer
\((-3,-1)\)

Solve the system by elimination.

\begin{align*} 5x+2y \amp = 5\\ 4x+3y \amp = -3 \end{align*}
Answer
\((3,-5)\)

Subsection 3. Solve a (special) 3x3 linear system

In this special case of solving a 3x3 system, we can eliminate \(c\) to create a 2x2 system.

Subsubsection Example

Example 6.94.

Solve the system by elimination.

\begin{align*} a+b+c \amp = 3 \amp \amp \text{(1)}\\ 4a-2b+c \amp = 18 \amp \amp \text{(2)}\\ 9a+3b+c \amp = 13 \amp \amp \text{(3)} \end{align*}
Solution

Eliminate \(c\) by subtracting (1) from (2), then eliminate \(c\) again by subtracting (1) from (3), to get a 2x2 system:

\begin{align*} 3a-3b \amp = 15 \\ 8a+2b \amp = 10 \end{align*}

Divide the first equation by 3 and the second equation by 2, then add.

\begin{align*} a-b \amp = 5 \\ \underline{4a+b} \amp \underline{{}= 5 \vphantom{4a+b}} \\ 5a \amp = 10 \end{align*}

We see that \(a=2\text{.}\) Substituting \(a=2\) into the equation \(a-b=5\text{,}\) we find that \(b=-3\text{.}\) Finally, we substitue \(a=2\) and \(b=-3\) into equation (1) to find

\begin{align*} 2-3+c \amp = 3 \\ c \amp = 4 \end{align*}

The solution is \(a=2,~b=-3\text{,}\) and \(c=4\text{.}\)

Subsubsection Exercise

Solve the system by elimination.

\begin{align*} a+b+c \amp = 5\\ 4a-2b+c \amp = -7\\ 16a+4b+c \amp = -37 \end{align*}
Answer
\(a=-3,~b=1,~c=7\)