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)
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.
The only expression that produces a quadratic function is ( ab ) ( x ) . ( 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 is a polynomial function of degree 2, which has the general form f ( x ) = c x 2 + d x + e , where c = 0 .
Analyzing Each Expression Let's examine each expression:
( ab ) ( x ) = a ( x ) ⋅ b ( x ) = ( 2 x − 4 ) ( x + 2 ) .
Expanding this product, 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 a term with x 2 and the coefficient of x 2 is non-zero (it is 2).
( b a ) ( x ) = b ( x ) a ( x ) = x + 2 2 x − 4 .
This is a rational function, not a quadratic function, because it is a ratio of two linear functions.
( a − b ) ( x ) = a ( x ) − b ( x ) = ( 2 x − 4 ) − ( x + 2 ) .
Simplifying this expression, we get: ( 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 ) .
Simplifying this expression, we get: ( 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 above analysis, only ( ab ) ( x ) = 2 x 2 − 8 results in a quadratic function.
Examples
Quadratic functions are incredibly useful in physics, engineering, and economics. For example, the trajectory of a projectile (like a ball thrown in the air) can be modeled by a quadratic function. Similarly, the shape of a satellite dish or a suspension bridge cable can be described using quadratic equations. In economics, quadratic functions can help model cost and revenue curves to optimize profit.
The expression that produces a quadratic function is (ab)(x), which results in the quadratic function 2x^2 - 8. The other options yield linear or rational functions but not quadratic functions.
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