Question 7: Consider the function defined by \(f(x)=cx^2+d\), where \(c\) and \(d\) are constant numbers. If \(f(1)=6\) and \(f(-2)=10\), then find the values of \(c\) and \(d\).
Solution:
\(f(x)=cx^2+d\)
\(f(1)=c{(1)}^2+d\)
\(c+d=6\)\(\ \_\_\_\_\_\_\ (1)\)\(\quad \because f(1)=6\)
\(f(-2)=c{(-2)}^2+d\)
\(4c+d=10\)\(\ \_\_\_\_\_\_\ (2)\)\(\quad \because f(-2)=10\)
Subtracting equation \((1)\) from \((2)\), we have
\(\ \ 4c+d=10\)
\(\underline{\underset{-}{}\ \ c\underset{-}{+}d=\underset{-}{}6}\)
\(\ \ 3c\ \ \ \ \ \ \ =4\)
\(c=\large{\frac{4}{3}}\)
Put value of \(c\) in equation \((1)\), we have
\(\large{\frac{4}{3}}+d=6\)
\(d=6-\large{\frac{4}{3}}\)
\(d=\large{\frac{18-4}{3}}\)
\(b=\large{\frac{14}{3}}\)