Single Variable Calculus, Chapter 1, 1.2, Section 1.2, Problem 8

The graphs shown below are a quadratic functions. Find expressions of each functions.













$\begin{equation} \begin{aligned} y &= ax^2 + bx + c, &&\text{general equation of quadratic function}\\ f(x) &= ax^2 + bx + c; (3,0), (4,2), (2,2)\\ 0 &= a(3)^2 + b(3) + c\\ 0 &= 9a + 3b + c &&\text{Equation 1}\\ 2 &= a(4)^2 + b(4) + c \\ 2 &= 16a + 4b + c &&\text{Equation 2}\\ 2 &= a(2) ^ 2 + b(2) + c\\ 2 &= 4a + 2b + c &&\text{Equation 3}\\ \end{aligned} \end{equation}$


*Combining Equations 1, 2 and 3 will result to
$a = 2 , b = -12, c = 18$
$\boxed{.: f(x) = 2x^2 - 12x + 18}$


$\begin{equation} \begin{aligned} g(x) &= ax^2 + bx + c; (-2, 2 ),(0, 1),(1, -2.5) \\ 2 &= a (2)^2 + b(-2) + c \\ 2 &= 4a - 2b + c && \text{Equation 1}\\ 1 &= a (0)^2 + b(0) + c\\ 1 &= c && \text{Equation 2}\\ -2.5 &= a(1)^2 + b(1) + c \\ -2.5 &= a + b + c && \text{Equation 3}\\ \end{aligned} \end{equation}$


*Combining Equations 1, 2 and 3 will result to
$a = -1, b = -2.5, c = 1$
$\boxed{.: g(x) = -x^2 - 2.5x + 1}$