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Chapter 5: Problem 21

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

Short Answer

Expert verified
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 by step solution

01

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: \(4a + 2b + c = 5\).
02

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.
03

Formulate Quadratic Function

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

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Most popular questions from this chapter

Find the partial fraction decomposition for \(\frac{1}{x(x+1)}\) and use the result to find the following sum: $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{99 \cdot 100} $$

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