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Chapter 5: Problem 22

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 quadratic function is: \(y = 4x^{2} + x - 7\)

Step by step solution

01

Substitute the points into the quadratic function

Start by substituting the given points into the equation for \(y\). This will create a system of three equations.\n1) Substituting the point \((-2,7)\) into the equation gives: \(7 = a(-2)^2 + b(-2) + c\). Simplify this equation to: \(7 = 4a - 2b + c\).\n2) Substituting the point \((1,-2)\) into the equation gives: \(-2 = a(1)^2 + b(1) + c\). Simplify this equation to: \(-2 = a + b + c\).\n3) Substituting the point \((2,3)\) into the equation gives: \(3 = a(2)^2 + b(2) + c\). Simplify this equation to: \(3 = 4a + 2b + c\).
02

Solve the system of equations

Now solve this system of linear equations for \(a\), \(b\), and \(c\). To make it easier, try to subtract second equation from the first and third equations: \n1) Subtracting the second equation from the first gives: \(9 = 3a - 3b\). Simplify it further to get \(a - b = 3\). \n2) Subtracting the second equation from the third gives: \(5 = 3a + b\).
03

Find the solution for a and b

Now we have system of two equations: \n1) \(a - b = 3\) \n2) \(a + b = 5\). Adding these two equations together gives \(2a = 8\) which simplifies to \(a = 4\). Substituting \(a = 4\) into the second equation gives \(b = 1\).
04

Find the value of c

Substitute \(a = 4\) and \(b = 1\) into one of the initial three equations to solve for \(c\). We will use equation \(-2 = a + b + c\), hence \(c = -2 - a - b = -2 - 4 - 1 = -7\).

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Most popular questions from this chapter

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$ y \leq 4 x+4 $$

Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125\). (In solving this exercise, let \(x\) represent the number of sold-out performances.)

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Exercises \(47-50\) describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\) c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000 .\) It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

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