Bushels of corn exported from the U.S., in millions, from $2008$ to $2012$ , can be modeled by a quadratic function. In $2008$ , the U.S. exported approximately $1849$ million bushels of corn. In $2009$ , the U.S. exported approximately $1979$ million bushels of corn, which was a maximum for that time period. According to the above information, which of the following best approximates the U.S. corn exports in $2012$ ? $\newline$ A. $809$ million bushels $\newline$ B. $1170$ million bushels $\newline$ C. $1589$ million bushels $\newline$ D. $2369$ million bushels

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Q. Bushels of corn exported from the U.S., in millions, from

2008

2012

, can be modeled by a quadratic function. In

2008

, the U.S. exported approximately

1849

million bushels of corn. In

2009

, the U.S. exported approximately

1979

million bushels of corn, which was a maximum for that time period. According to the above information, which of the following best approximates the U.S. corn exports in

2012

\newline

809

million bushels

\newline

1170

million bushels

\newline

1589

million bushels

\newline

2369

million bushels

Understand given information: Understand the given information and what is being asked. We are given that the U.S. corn exports can be modeled by a quadratic function. We know the exports for $2008$ and that $2009$ was a maximum. We need to find the exports for $2012$ .
Recognize quadratic function form: Recognize that a quadratic function has the general form $y = ax^2 + bx + c$ . Since $2009$ is a maximum, the graph of the function opens downwards, and the vertex of the parabola will be at the point $(2009, 1979)$ . We also know the point $(2008, 1849)$ lies on the parabola.
Use vertex form of quadratic function: Use the vertex form of a quadratic function, which is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. In this case, $h = 2009$ and $k = 1979$ . We can substitute these values into the vertex form to get $y = a(x - 2009)^2 + 1979$ .
Find value of $a$ : Use the point $(2008, 1849)$ to find the value of $a$ . Substituting $x = 2008$ and $y = 1849$ into the equation from Step $3$ gives us $1849 = a(2008 - 2009)^2 + 1979$ . Simplifying, we get $-130 = a(-1)^2$ , so $a = -130$ .
Write complete quadratic function: Now that we have the value of $a$ , we can write the complete quadratic function as $y = -130(x - 2009)^2 + 1979$ .
Substitute $x = 2012$ : To find the exports in $2012$ , substitute $x = 2012$ into the quadratic function. This gives us $y = -130(2012 - 2009)^2 + 1979$ . Simplifying, we get $y = -130(3)^2 + 1979$ , which is $y = -130(9) + 1979$ .
Calculate y value: Calculate the value of y. Multiplying $-130$ by $9$ gives $-1170$ . Adding this to $1979$ gives us $y = 1979 - 1170$ , which is $y = 809$ .
Compare with given options: Compare the calculated value of $y$ with the given options. The value we calculated, $809$ million bushels, matches option $A$ .