The graph of which function has an axis of symmetry at [tex]$x=-\frac{1}{4} ?$[/tex]
[tex]$f(x)=2 x^2+x-1$[/tex]
[tex]$f(x)=2 x^2-x+1$[/tex]
[tex]$f(x)=x^2+2 x-1$[/tex]
[tex]$f(x)=x^2-2 x+1$[/tex]
Answer (2)
The axis of symmetry of a quadratic function f ( x ) = a x 2 + b x + c is given by x = − 2 a b .
Calculate the axis of symmetry for each given function.
f ( x ) = 2 x 2 + x − 1 has an axis of symmetry at x = − 4 1 .
The function with the axis of symmetry at x = − 4 1 is f ( x ) = 2 x 2 + x − 1 .
Explanation
Understanding the Problem We are given four quadratic functions and we need to find the one whose axis of symmetry is at x = − 4 1 . The axis of symmetry for a quadratic function in the form f ( x ) = a x 2 + b x + c is given by the formula x = − 2 a b . We will apply this formula to each of the given functions.
Calculating Axis of Symmetry for the First Function For f ( x ) = 2 x 2 + x − 1 , we have a = 2 and b = 1 . The axis of symmetry is x = − 2 ( 2 ) 1 = − 4 1 .
Calculating Axis of Symmetry for the Second Function For f ( x ) = 2 x 2 − x + 1 , we have a = 2 and b = − 1 . The axis of symmetry is x = − 2 ( 2 ) − 1 = 4 1 .
Calculating Axis of Symmetry for the Third Function For f ( x ) = x 2 + 2 x − 1 , we have a = 1 and b = 2 . The axis of symmetry is x = − 2 ( 1 ) 2 = − 1 .
Calculating Axis of Symmetry for the Fourth Function For f ( x ) = x 2 − 2 x + 1 , we have a = 1 and b = − 2 . The axis of symmetry is x = − 2 ( 1 ) − 2 = 1 .
Finding the Matching Function Comparing the calculated axes of symmetry with the given axis of symmetry x = − 4 1 , we find that the function f ( x ) = 2 x 2 + x − 1 has the axis of symmetry at x = − 4 1 .
Examples
Understanding the axis of symmetry is crucial in various fields, such as physics and engineering. For instance, when designing a parabolic reflector for a satellite dish, knowing the axis of symmetry helps in positioning the receiver at the focus to maximize signal reception. Similarly, in architecture, symmetrical structures often distribute weight evenly, enhancing stability. By finding the axis of symmetry, we can optimize designs and ensure balance and efficiency.
The function with the axis of symmetry at x = − 4 1 is f ( x ) = 2 x 2 + x − 1 . This is determined by applying the formula for the axis of symmetry of quadratic functions. The other functions do not match this value.
;
