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Chapter 5: Problem 20

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-2,7),(1,-2),(2,3)$$

Short Answer

Expert verified
The process involves creating a system of equations with the given points and the quadratic function, then solving the system to find the coefficients of the function. The solution is the quadratic function \(y = ax^{2} + bx + c\) with the calculated coefficients.

Step by step solution

01

Formulate the system of equations

The equation of the curve is given by \(y = ax^2 + bx + c\). The three points (-2,7), (1, -2), and (2,3) should satisfy this equation, which gives us a system of three equations as follows: \n Equation 1: \(a*(-2)^2 + b*(-2) + c = 7\)\n Equation 2: \(a*1^2 + b*1 + c = -2\)\n Equation 3: \(a*2^2 + b*2 + c = 3\)
02

Solve the first equation to simplify the system

Solving the first equation results in \(4a - 2b + c = 7\). This helps simplify the system of equations.
03

Substitute the coefficients in the remaining equations

Substitute the simplified form of equation 1 into the other two equations, we get a smaller system of equations:\n From equation 2: \(a + b + c = -2\)\n and from equation 3: \(4a + 2b + c = 3\).
04

Solve the system of equations

This system of equations can now be solved using substitution or elimination methods. These equations are now enough to solve for a, b, and c.
05

Write down the quadratic function

Once the values of a, b and c are found, they can be substituted back into \(y = ax^{2} + bx + c\) to find the quadratic equation that passes through the given points.

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Most popular questions from this chapter

write the partial fraction decomposition of each rational expression. $$ \frac{3 x^{2}+49}{x(x+7)^{2}} $$

How can you verify your result for the partial fraction decomposition for a given rational expression without using a graphing utility?

Consider the following array of numbers:$$\left[\begin{array}{ccc}1 & 2 & -1 \\\ 4 & -3 &-15\end{array}\right]$$ Rewrite the array as follows: Multiply each number in the top row by \(-4\) and add this product to the corresponding number in the bottom row. Do not change the numbers in the top row.

write the partial fraction decomposition of each rational expression. $$ \frac{1}{x^{2}-a x-b x+a b} \quad(a \neq b) $$

will help you prepare for the material covered in the next section. Solve by the substitution method: $$ \left\\{\begin{array}{l} 4 x+3 y=4 \\ y=2 x-7 \end{array}\right. $$
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