Which quadratic function in standard form has the values [tex]$a=-3.5, b=2.7$[/tex], and [tex]$c=-8.2$[/tex]?
A. [tex]$f(x)=-3.5 x^2-8.2 x+2.7$[/tex]
B. [tex]$f(x)=2.7 x^2-8.2 x-3.5$[/tex]
C. [tex]$f(x)=-3.5 x^2+2.7 x-8.2$[/tex]
D. [tex]$f(x)=2.7 x^2-3.5 x-8.2$[/tex]
Answer (2)
Substitute the given values a = − 3.5 , b = 2.7 , and c = − 8.2 into the standard form of a quadratic function f ( x ) = a x 2 + b x + c .
Obtain the quadratic function f ( x ) = − 3.5 x 2 + 2.7 x − 8.2 .
The quadratic function in standard form is f ( x ) = − 3.5 x 2 + 2.7 x − 8.2 .
Explanation
Understanding the Problem We are given the values a = − 3.5 , b = 2.7 , and c = − 8.2 for a quadratic function in standard form, which is f ( x ) = a x 2 + b x + c . Our goal is to substitute these values into the standard form to find the correct quadratic function.
Substituting the Values Substitute the given values into the standard form: f ( x ) = a x 2 + b x + c = − 3.5 x 2 + 2.7 x + ( − 8.2 ) = − 3.5 x 2 + 2.7 x − 8.2
Final Answer The quadratic function in standard form with the given values is f ( x ) = − 3.5 x 2 + 2.7 x − 8.2 .
Examples
Quadratic functions are used in various real-world applications, such as modeling the trajectory of a projectile, designing parabolic mirrors, or determining the optimal dimensions for a rectangular enclosure. For example, if you want to model the height of a ball thrown into the air, you can use a quadratic function where 'a' represents the effect of gravity, 'b' represents the initial upward velocity, and 'c' represents the initial height from which the ball was thrown. By understanding the coefficients of the quadratic function, you can predict the ball's maximum height and how long it will stay in the air.
The quadratic function in standard form with the coefficients provided is option C: f ( x ) = − 3.5 x 2 + 2.7 x − 8.2 . By substituting the values of a, b, and c into the standard form f ( x ) = a x 2 + b x + c , we get the function. This shows how to correctly identify the appropriate quadratic function from multiple choices.
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