Consider the function f(x)=56×2. Part A
What type of function does the equation model?
A. Linear
B. Quadratic
C. Exponential
D. Absolute value
Part B
What is the value of the function when x = 12?
The Correct Answer and Explanation is:
To analyze the function ( f(x) = 56x^2 ), we will go through each part of your question.
Part A: Type of Function
The function ( f(x) = 56x^2 ) is a quadratic function.
- Linear Functions are characterized by a constant rate of change and can be represented in the form ( f(x) = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. Linear functions produce a straight line when graphed.
- Quadratic Functions, like our given function, have the general form ( f(x) = ax^2 + bx + c ) where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ). The presence of the ( x^2 ) term indicates that the function will produce a parabolic shape when graphed. In this case, ( a = 56 ) is a positive constant, which means the parabola opens upwards.
- Exponential Functions can be expressed in the form ( f(x) = a \cdot b^x ), where ( a ) is a constant, and ( b ) is the base of the exponent. These functions grow much faster than quadratic functions.
- Absolute Value Functions are represented as ( f(x) = |x| ) and display a distinct V-shape on a graph, with the output being always non-negative.
Given that ( f(x) = 56x^2 ) contains the term ( x^2 ) and matches the form of a quadratic function, the correct answer is B. Quadratic.
Part B: Value of the Function at ( x = 12 )
To find the value of the function when ( x = 12 ), we substitute ( x ) into the function:
[
f(12) = 56(12^2)
]
Calculating ( 12^2 ):
[
12^2 = 144
]
Now substitute ( 144 ) back into the function:
[
f(12) = 56 \cdot 144
]
To perform this multiplication:
[
56 \cdot 144 = 8064
]
Thus, the value of the function at ( x = 12 ) is:
[
\boxed{8064}
]
Explanation
The evaluation of ( f(12) ) demonstrates how quadratic functions behave as the input values increase. As seen in the calculations, the output grows significantly as ( x ) becomes larger. This characteristic highlights the non-linear growth of quadratic functions compared to linear functions. Quadratics can model various real-world phenomena where acceleration or growth is involved, such as projectile motion or certain financial calculations. The ( x^2 ) term leads to exponential-like increases in values as ( x ) rises, which is evident in our calculation where ( f(12) ) yields 8064. Such functions are crucial in fields like physics, engineering, and economics, providing insights into relationships involving squared terms, thus validating their application in real-world scenarios.