Which of the following functions is not quadratic?
[tex]$f(x)=2 x^2+5 x-1$[/tex]
[tex]$f^{\prime}(x)=-3 x^2-1$[/tex]
[tex]$f(x)=5 x-5$[/tex]
[tex]$f^{\prime}(x)=x^2$[/tex]
Answer (1)
A quadratic function has the form f ( x ) = a x 2 + b x + c , where a e q 0 .
Examine each function to determine if it fits the quadratic form.
f ( x ) = 5 x − 5 is a linear function, not a quadratic function.
The function that is not quadratic is f ( x ) = 5 x − 5 .
Explanation
Understanding Quadratic Functions A quadratic function is a polynomial function of degree 2. The general form of a quadratic function is f ( x ) = a x 2 + b x + c , where a e q 0 . We need to identify which of the given functions is not quadratic.
Analyzing Each Function Let's examine each function:
f ( x ) = 2 x 2 + 5 x − 1 : This is a quadratic function because it has a term with x 2 and fits the general form a x 2 + b x + c where a = 2 , b = 5 , and c = − 1 .
f ′ ( x ) = − 3 x 2 − 1 : This is also a quadratic function because it has a term with x 2 and fits the general form a x 2 + b x + c where a = − 3 , b = 0 , and c = − 1 .
f ( x ) = 5 x − 5 : This is a linear function (a polynomial of degree 1) because the highest power of x is 1. It fits the general form of a linear function f ( x ) = m x + b where m = 5 and b = − 5 . Therefore, it is not a quadratic function.
f ′ ( x ) = x 2 : This is a quadratic function because it has a term with x 2 and fits the general form a x 2 + b x + c where a = 1 , b = 0 , and c = 0 .
Identifying the Non-Quadratic Function The function that is not quadratic is f ( x ) = 5 x − 5 .
Examples
Understanding quadratic functions helps in modeling various real-world scenarios, such as the trajectory of a projectile, the shape of a satellite dish, or the design of arched bridges. For instance, if you throw a ball, its path through the air can be approximated by a quadratic function, allowing you to predict where it will land. Similarly, engineers use quadratic functions to optimize the design of curved structures, ensuring stability and efficiency.
