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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. $$(-1,6),(1,4),(2,9)$$

Short Answer

Expert verified
The quadratic function that passes through the points (-1,6), (1,4), and (2,9) is \(y = 3x^2 - 2x + 5\).

Step by step solution

01

Formulate Systems of Equations

Substitute each given point into the general form of the quadratic equation \(y = ax^2 + bx + c\).From (-1,6): \[6 = a(-1)^2 - b + c\]From (1,4):\[4 = a + b + c\]From (2,9): \[9 = 4a + 2b + c\]
02

Solve the System of Equations

You now have the system of equations:\[\begin{align*}6 &= a - b + c,\4 &= a + b + c,\9 &= 4a + 2b + c.\end{align*}\]Subtract the first equation from the second to find \(b = -2\). Substitute \(b = -2\) into the first two equations to simultaneously solve the equations and find \(a = 3\) and \(c = 5\) respectively.
03

Write down the Quadratic Function

Substitute \(a = 3\), \(b = -2\), and \(c = 5\) into the standard form of a quadratic equation \(y = ax^2 + bx + c\). This gives:\[y = 3x^2 - 2x + 5\].

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Most popular questions from this chapter

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