Determine whether each quadratic function is an even function. answer yes or no
The Correct Answer and Explanation is :
To determine if a quadratic function is an even function, we need to examine its definition. A function ( f(x) ) is considered even if it satisfies the condition ( f(-x) = f(x) ) for all values of ( x ) in its domain.
Quadratic functions are generally expressed in the form:
[
f(x) = ax^2 + bx + c
]
where ( a ), ( b ), and ( c ) are constants. To determine if a quadratic function is even, we can follow these steps:
- Evaluate ( f(-x) ): Substitute (-x) into the function.
[
f(-x) = a(-x)^2 + b(-x) + c = ax^2 – bx + c
] - Compare ( f(-x) ) to ( f(x) ): We then compare ( f(-x) ) to ( f(x) ).
[
f(x) = ax^2 + bx + c
] - Check the condition: For the function to be even, ( f(-x) ) must equal ( f(x) ). Setting them equal gives:
[
ax^2 – bx + c = ax^2 + bx + c
]
Simplifying this, we can cancel out ( ax^2 ) and ( c ) on both sides, leading to:
[
-bx = bx
]
This implies:
[
-2bx = 0
]
This is only true for all ( x ) if ( b = 0 ).
Conclusion: A quadratic function is even if it does not have a linear term, meaning ( b = 0 ). The function will then simplify to ( f(x) = ax^2 + c ), which satisfies the even function condition, as it is symmetrical about the y-axis.
Examples:
- ( f(x) = 2x^2 + 3 ) (Yes, even)
- ( f(x) = x^2 – 4x + 1 ) (No, not even)
So, the answer for whether a quadratic function is an even function is “yes” if ( b = 0 ) and “no” otherwise.