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Chapter 5: Problem 24

Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(1,3),(3,-1),(4,0)$$

Short Answer

Expert verified
The quadratic function is \(y = x^2 - 4x + 6\).

Step by step solution

01

Formation of Equations

The quadratic function is defined as \(y = ax^2 + bx + c\). By inserting the x and y values from the three points given, the following equations are obtained: \From \((1,3)\), we get \(3 = a + b + c\),\From \((3,-1)\), we get \(-1 = 9a + 3b + c\),\From \((4,0)\), we get \(0 = 16a + 4b + c\).
02

Solving System of Equations

By subtracting the first equation from the second and third equations, the system is simpler :\After subtraction, two new equations are obtained: \\(9a + 3b - a - b = -4 \Rightarrow 8a + 2b = -4 \), and \\(16a + 4b - a - b = -3 \Rightarrow 15a + 3b = -3 \).\These are easier to solve. Dividing the first one by 2 and the second one by 3 gives: \\(4a + b = -2\), and \\(5a + b = -1\).\Subtracting these two gives \(-a = -1 \Rightarrow a = 1 \). Inserting \(a = 1\) into \(4a + b = -2 \) gives \(b = -4\). Inserting \(a = 1\) and \(b = - 4\) into the first original equation \(3 = a + b + c \) gives \(c = 6\).
03

Finding the Quadratic Function

Inserting \(a = 1\), \(b = -4\), and \(c = 6\) back into the original equation \(y = ax^2 + bx + c\) gives the specific quadratic function \(y = x^2 - 4x + 6\).

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Most popular questions from this chapter

In \(1985,\) college graduates averaged \(\$ 508\) in weekly earnings. This amount has increased by approximately \(\$ 25\) in weekly earnings per year. By contrast, in 1985 , people with less than four years of high school averaged \(\$ 270\) in weekly earnings. This amount has only increased by approximately \(\$ 4\) in weekly earnings per year. a. Write a function that models weekly earnings, \(E,\) for college graduates \(x\) years after 1985 b. Write a function that models weekly earnings, \(E,\) for people with less than four years of high school \(x\) years after 1985 c. How many years after 1985 will college graduates be earning three times as much per week as people with less than four years of high school? (Round to the nearest whole number.) In which year will this occur? What will be the weekly earnings for each group at that time? (GRAPH CAN'T COPY)

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=6\) \(x-y=-2\)

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=6\) \(2 x-3 y=6\)

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.)

A person with no more than 15,000 dollars to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least 2000 dollars is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high- and low-risk investments.
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