In the following function defined by an equation in the form [tex]$y=a x^2+b x+c$[/tex], identify the values of [tex]$a, b$[/tex], and [tex]$c$[/tex]. [tex]$y=-2 x^2$[/tex]
a. [tex]$a=-2, b=0, c=0$[/tex]
b. [tex]$a=-2, b=-2, c=-2$[/tex]
c. [tex]$a=0, b=-2, c=0$[/tex]
d. [tex]$a=0, b=0, c=-2$[/tex]
Please select the best answer from the choices provided
Answer (1)
Compare the given equation y = − 2 x 2 with the general form y = a x 2 + b x + c .
Identify the coefficient of the x 2 term as the value of a : a = − 2 .
Identify the coefficient of the x term as the value of b : b = 0 .
Identify the constant term as the value of c : c = 0 . The final answer is a = − 2 , b = 0 , c = 0 .
Explanation
Problem Analysis We are given the equation y = − 2 x 2 and we want to find the values of a , b , and c such that y = a x 2 + b x + c .
Comparing Equations Comparing y = − 2 x 2 with y = a x 2 + b x + c , we can identify the coefficients of each term.
Finding a The coefficient of the x 2 term in y = − 2 x 2 is − 2 . Therefore, a = − 2 .
Finding b The coefficient of the x term in y = − 2 x 2 is 0 because there is no x term. Therefore, b = 0 .
Finding c The constant term in y = − 2 x 2 is 0 because there is no constant term. Therefore, c = 0 .
Final Values Thus, we have a = − 2 , b = 0 , and c = 0 .
Examples
Understanding quadratic functions is crucial in various fields, such as physics and engineering. For instance, the trajectory of a projectile under constant gravitational acceleration can be modeled by a quadratic function. By identifying the coefficients a , b , and c , we can determine key characteristics of the projectile's path, such as its maximum height and range. This knowledge is essential for designing accurate targeting systems and predicting the behavior of objects in motion.
