Select the correct answer.
The zeros of a quadratic function are 6 and -4. Which of these choices could be the function?
A. [tex]$f(x)=(x+6)(x-4)$[/tex]
B. [tex]$f(x)=(x+6)(x+4)$[/tex]
C. [tex]$f(x)=(x-6)(x+4)$[/tex]
D. [tex]$f(x)=(x-6)(x-4)$[/tex]
Answer (1)
The zeros of the quadratic function are 6 and -4.
A quadratic function with zeros at x = a and x = b can be written as f ( x ) = ( x − a ) ( x − b ) .
Substituting the given zeros, we get f ( x ) = ( x − 6 ) ( x + 4 ) .
The correct answer is f ( x ) = ( x − 6 ) ( x + 4 ) .
Explanation
Understanding the Zeros We are given that the zeros of a quadratic function are 6 and -4. This means that the function equals zero when x = 6 and x = − 4 . We need to find the quadratic function that has these zeros.
Forming the Quadratic Function A quadratic function with zeros at x = a and x = b can be written in the form f ( x ) = ( x − a ) ( x − b ) . This is because when x = a or x = b , the function becomes zero.
Substituting the Zeros In our case, the zeros are a = 6 and b = − 4 . Substituting these values into the expression, we get:
f ( x ) = ( x − 6 ) ( x − ( − 4 )) = ( x − 6 ) ( x + 4 )
This is because when x = 6 , f ( 6 ) = ( 6 − 6 ) ( 6 + 4 ) = 0 × 10 = 0 , and when x = − 4 , f ( − 4 ) = ( − 4 − 6 ) ( − 4 + 4 ) = − 10 × 0 = 0 .
Identifying the Correct Function Now, we compare our result f ( x ) = ( x − 6 ) ( x + 4 ) with the given choices:
A. f ( x ) = ( x + 6 ) ( x − 4 ) B. f ( x ) = ( x + 6 ) ( x + 4 ) C. f ( x ) = ( x − 6 ) ( x + 4 ) D. f ( x ) = ( x − 6 ) ( x − 4 )
We can see that choice C matches our result.
Final Answer Therefore, the correct answer is C. f ( x ) = ( x − 6 ) ( x + 4 ) .
Examples
Understanding quadratic functions and their zeros is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight, knowing the zeros helps determine the optimal placement of the light source to maximize the beam's intensity. Similarly, in projectile motion, the zeros of the quadratic function representing the projectile's height can tell us when the projectile hits the ground. This knowledge is essential in fields like engineering, physics, and sports science.
