Determine whether the function is linear or quadratic. Identify the quadratic, linear, and constant terms.
[tex]f(x)=(3 x-4)(-6 x+5)[/tex]
Answer (1)
Expand the function: f ( x ) = ( 3 x − 4 ) ( − 6 x + 5 ) = − 18 x 2 + 39 x − 20 .
Identify the type of function: Since the highest power of x is 2, the function is quadratic.
Identify the quadratic, linear, and constant terms: Quadratic term: − 18 x 2 , Linear term: 39 x , Constant term: − 20 .
State the final answer: The function is quadratic with terms − 18 x 2 , 39 x , and − 20 . Quadratic function, Quadratic term: − 18 x 2 , Linear term: 39 x , Constant term: − 20
Explanation
Expanding the Function First, we need to expand the given function f ( x ) = ( 3 x − 4 ) ( − 6 x + 5 ) to determine if it is linear or quadratic.
Expanding the Expression Expanding the expression, we get:
f ( x ) = ( 3 x − 4 ) ( − 6 x + 5 ) = 3 x ( − 6 x + 5 ) − 4 ( − 6 x + 5 ) = − 18 x 2 + 15 x + 24 x − 20 = − 18 x 2 + 39 x − 20
Identifying the Terms Now, we identify the terms. The highest power of x is 2, so the function is quadratic. The quadratic term is the coefficient of x 2 , which is − 18 x 2 . The linear term is the coefficient of x , which is 39 x . The constant term is − 20 .
Conclusion Therefore, the function is quadratic, with a quadratic term of − 18 x 2 , a linear term of 39 x , and a constant term of − 20 .
Examples
Understanding whether a function is linear or quadratic is crucial in many real-world applications. For example, the path of a ball thrown in the air can be modeled by a quadratic function, where the height of the ball depends on time. Similarly, the trajectory of a rocket or the shape of a satellite dish can be described using quadratic functions. Recognizing these functions helps in predicting outcomes and designing systems efficiently.
