Consider the quadratic function.
[tex]f(p)=p^2-8 p-5[/tex]
What are the values of the coefficients and the constant in the function?
A. [tex]a=-1, b=-8, c=-5[/tex]
B. [tex]a=1, b=-5, c=-8[/tex]
C. [tex]a=1, b=-8, c=-5[/tex]
D. [tex]a=-1, b=-5, c=8[/tex]
Answer (1)
Identify the coefficient of p 2 as a , which is 1 .
Identify the coefficient of p as b , which is − 8 .
Identify the constant term as c , which is − 5 .
State the values: a = 1 , b = − 8 , c = − 5 .
Explanation
Understanding the Quadratic Function We are given the quadratic function f ( p ) = p 2 − 8 p − 5 . Our goal is to identify the coefficients a , b , and the constant term c in this function.
General Form of a Quadratic Function The general form of a quadratic function is f ( p ) = a p 2 + b p + c , where:
a is the coefficient of the p 2 term,
b is the coefficient of the p term,
c is the constant term.
Identifying the Coefficients and Constant Term Comparing the given function f ( p ) = p 2 − 8 p − 5 with the general form f ( p ) = a p 2 + b p + c , we can identify the values of a , b , and c :
The coefficient of p 2 is 1 , so a = 1 .
The coefficient of p is − 8 , so b = − 8 .
The constant term is − 5 , so c = − 5 .
Final Values Therefore, the values of the coefficients and the constant term are a = 1 , b = − 8 , and c = − 5 .
Examples
Quadratic functions are used in various real-life applications, such as modeling the trajectory of a projectile, designing parabolic mirrors and reflectors, and determining the minimum or maximum values in optimization problems. For example, if you throw a ball, its path can be modeled by a quadratic function, where the coefficients determine the initial velocity and angle of the throw. By understanding the coefficients, you can predict how far the ball will travel and how high it will go.
