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Chapter 9: Problem 39

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-2)=-4, f(1)=2,\) and \(f(2)=0\)

Short Answer

Expert verified
The quadratic function is \(f(x)= -x^{2} + x + 2\).

Step by step solution

01

Formulate the System of Equations

Using the given points, three equations can be formed by replacing x and f(x) in the function \(f(x)=a x^{2}+b x+c\):1) For the first point \(-2,-4\), we get \(a(-2)^2 + b(-2) + c = -4 \)2) For the second point \(1,2\), we get \(a(1)^2 + b(1) + c = 2\)3) For the third point \(2,0\), we get \(a(2)^2 + b(2) + c = 0\)
02

Simplify the Equations

We simplify above equations to:1) \(4a - 2b + c = -4\)2) \(a + b + c = 2\)3) \(4a + 2b + c = 0\)
03

Solve the System of Equations

Solve this system of equations using a methodology such as elimination or substitution. For simplicity, we will add first and third equation and substract the second equation from the result. This gives us a new equation that is: \(a= -1\). Now, we will substitute \(a= -1\) into the second and third equations to find b and c.
04

Find a, b, and c

By substituting \(-1\) for \(a\) into the second equation, we find \(b=1\) and by substituting both \(-1\) for \(a\) and \(1\) for \(b\) into the third equation, we find \(c= 2\).
05

Substitute a, b, and c into the Quadratic Function

Substitute \(a= -1\), \(b= 1\), and \(c= 2\) into the quadratic function \(f(x)= a x^{2}+b x+c\). This yields the function \(f(x)= -x^{2} + x + 2\) as the final output.

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Most popular questions from this chapter

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 3 a+b-c &=0 \\ 2 a+3 b-5 c &=1 \\ a-2 b+3 c &=-4 \end{aligned}\right. $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{l} {2 x+2 y+7 z=-1} \\ {2 x+y+2 z=2} \\ {4 x+6 y+z=15} \end{array}\right. $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{r} {2} \\ {-4} \\ {1} \end{array}\right], \quad B=\left[\begin{array}{r} {-5} \\ {3} \\ {-1} \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ 4 A+3 B=-2 X $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with two matrices that can be multiplied but not added.
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