Problem 21 - College Algebra Essentials

Question

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-1,-4),(1,-2),(2,5)$$

Step-by-Step Solution

Answer

Without full calculations, the coefficients of quadratic function cannot determined. Solve the formed linear system and substitute those values back into the quadratic function to get the final function.

Step 1: Formulate Equations

Using the quadratic function's form \(y=ax^{2}+bx+c\), replace \(x\) and \(y\) with the values provided from each point to form three equations.\n 1. For (-1, -4) the equation is: \(a*(-1)^2 + b*(-1) + c = -4\), which simplifies to: \(a - b + c = -4\).2. For (1, -2) the equation is: \(a*(1)^2 + b*(1) + c = -2\), which simplifies to: \(a + b + c = -2\).3. For (2, 5) the equation is: \(a*(2)^2 + b*(2) + c = 5\), which simplifies to: \(4a + 2b + c = 5\).

Step 2: Solve Linear System

We now have a linear system of equations, which we can represent in matrix form as: \[ \left[ \begin{array}{ccc} 1 & -1 & 1 \ 1 & 1 & 1 \ 4 & 2 & 1 \end{array} \right] \left[ \begin{array}{c} a \ b \ c \end{array} \right] = \left[ \begin{array}{c} -4 \ -2 \ 5 \end{array} \right] \]. Solve this system of equations for \(a\), \(b\), and \(c\) using your method of choice, such as substitution, elimination or using a tool like a calculator.

Step 3: Formulate Quadratic Function

Finally, take found values \(a\), \(b\), and \(c\) and substitute them back into the general quadratic equation \(y=ax^{2}+bx+c\).