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Chapter 9: Problem 40

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-1)=5, f(1)=3,\) and \(f(2)=5\)

Short Answer

Expert verified
The quadratic function satisfying the given conditions is \(f(x) = x^2 - x + 3\).

Step by step solution

01

Set up system of equations

Using the expressions given \(f(-1)=5, f(1)=3,\) and \(f(2)=5\), create three equations. Plug \(-1\), \(1\), and \(2\) into the quadratic function \(f(x)=a*x^2+b*x+c\) and equate them to \(5\), \(3\), and \(5\) respectively. The resulting equations are: \[ a - b + c = 5 \] \[ a + b + c = 3 \] \[ 4a + 2b + c = 5 \]
02

Solve system of equations

Solve these three equations simultaneously using any of the methods for solving linear equations. Subtraction of the first two equations gives \(2b = -2\), which can be simplified to \(b = -1\). Replacing \(b\) with \(-1\) in the first equation leads to \(a - (-1) + c = 5\), which simplifies to \(a + c = 4\). Using \(b = -1\) in the third equation yields \(4a - 2 + c = 5\), which can be further simplified to \(4a + c = 7\). Thus the system of equations to solve is now: \[ a + c = 4 \] \[ 4a + c = 7 \]
03

Find a, b, and c

Solving the last two equations gives \(a = 1\) and \(c = 3\). Thus, the original function \(f(x) = ax^2 + bx + c\) with \(a = 1\), \(b = -1\), and \(c = 3\), can be expressed as: \[f(x) = x^2 - x + 3\].

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Most popular questions from this chapter

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{lll} {6} & {2} & {-3} \end{array}\right], B=\left[\begin{array}{lll} {4} & {-2} & {3} \end{array}\right] $$

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {2} & {4} \\ {3} & {1} \\ {4} & {2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {3} & {2} & {0} \\ {-1} & {-3} & {5} \end{array}\right] $$

Solve the system: $$ \left\\{\begin{array}{c} {2 \ln w+\ln x+3 \ln y-2 \ln z=-6} \\ {4 \ln w+3 \ln x+\ln y-\ln z=-2} \\ {\ln w+\ln x+\ln y+\ln z=-5} \\ {\ln w+\ln x-\ln y-\ln z=5} \end{array}\right. $$ (Hint: Let \(A=\ln w, B=\ln x, C=\ln y,\) and \(D=\ln z .\) Solve the system for \(A, B, C,\) and \(D .\) Then use the logarithmic equations to find \(w, x, y, \text { and } z .)\)

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} w+x+y+z &=4 \\ 2 w+x-2 y-z &=0 \\ w-2 x-y-2 z &=-2 \\ 3 w+2 x+y+3 z &=4 \end{aligned}\right. $$

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. $$ X-A=B $$
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