If $a(x)=2 x-4$ and $b(x)=x+2$, which of the following expressions produces a quadratic function?
A. $(a b)(x)$
B. $\left(\frac{a}{b}\right)(x)$
C. $(a-b)(x)$
D. $(a+b)(x)$
Answer (2)
The expression that produces a quadratic function is ( ab ) ( x ) , which equals 2 x 2 − 8 . All other expressions result in either linear or rational functions. Therefore, the correct choice is option A.
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Multiply a ( x ) and b ( x ) : ( ab ) ( x ) = ( 2 x − 4 ) ( x + 2 ) = 2 x 2 − 8 , which is a quadratic function.
Divide a ( x ) by b ( x ) : ( b a ) ( x ) = x + 2 2 x − 4 , which is a rational function.
Subtract b ( x ) from a ( x ) : ( a − b ) ( x ) = ( 2 x − 4 ) − ( x + 2 ) = x − 6 , which is a linear function.
Add a ( x ) and b ( x ) : ( a + b ) ( x ) = ( 2 x − 4 ) + ( x + 2 ) = 3 x − 2 , which is a linear function.
Therefore, the expression that produces a quadratic function is ( ab ) ( x ) .
Explanation
Understanding the Problem We are given two functions, a ( x ) = 2 x − 4 and b ( x ) = x + 2 . We need to determine which of the following expressions results in a quadratic function: ( ab ) ( x ) , ( b a ) ( x ) , ( a − b ) ( x ) , ( a + b ) ( x ) . A quadratic function has the form f ( x ) = A x 2 + B x + C , where A = 0 .
Analyzing Each Expression Let's analyze each expression:
( ab ) ( x ) = a ( x ) b ( x ) = ( 2 x − 4 ) ( x + 2 ) . Expanding this, we get: ( 2 x − 4 ) ( x + 2 ) = 2 x 2 + 4 x − 4 x − 8 = 2 x 2 − 8 This is a quadratic function because it has the form A x 2 + B x + C with A = 2 , B = 0 , and C = − 8 .
( b a ) ( x ) = b ( x ) a ( x ) = x + 2 2 x − 4 . This is a rational function, not a quadratic function.
( a − b ) ( x ) = a ( x ) − b ( x ) = ( 2 x − 4 ) − ( x + 2 ) = 2 x − 4 − x − 2 = x − 6 . This is a linear function, not a quadratic function.
( a + b ) ( x ) = a ( x ) + b ( x ) = ( 2 x − 4 ) + ( x + 2 ) = 2 x − 4 + x + 2 = 3 x − 2 . This is a linear function, not a quadratic function.
Conclusion From the analysis above, only ( ab ) ( x ) = 2 x 2 − 8 results in a quadratic function.
Examples
Quadratic functions are incredibly useful in real-world applications. For instance, the trajectory of a ball thrown into the air can be modeled by a quadratic function. Similarly, the shape of a satellite dish or the cross-section of a bridge's arch often follows a quadratic curve. Understanding quadratic functions helps engineers design structures, predict motion, and optimize various systems.
