Consider the function [tex]f(x)=-3 x^2+30 x-1[/tex].
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
a. The function has a $\square$ value.
Answer (2)
Determine that the function has a maximum value since the coefficient of the x 2 term is negative.
Calculate the x-coordinate of the vertex using the formula x = − 2 a b , which gives x = 5 .
Substitute x = 5 into the function to find the maximum value, which is f ( 5 ) = 74 .
Identify the domain as ( − ∞ , ∞ ) and the range as ( − ∞ , 74 ] .
ma x im u m
Explanation
Problem Analysis We are given the quadratic function f ( x ) = − 3 x 2 + 30 x − 1 . Our goal is to determine whether it has a minimum or maximum value, find that value and where it occurs, and identify the function's domain and range.
Determining Minimum or Maximum The function is a quadratic function in the form f ( x ) = a x 2 + b x + c , where a = − 3 , b = 30 , and c = − 1 . Since a = − 3 is negative, the parabola opens downward, which means the function has a maximum value.
Finding the x-coordinate of the vertex To find the x-coordinate of the vertex, we use the formula x = − 2 a b . Substituting the values of a and b , we get x = − 2 ( − 3 ) 30 = − − 6 30 = 5
Finding the maximum value To find the maximum value (y-coordinate of the vertex), we substitute x = 5 into the function: f ( 5 ) = − 3 ( 5 ) 2 + 30 ( 5 ) − 1 = − 3 ( 25 ) + 150 − 1 = − 75 + 150 − 1 = 74 So, the maximum value is 74, and it occurs at x = 5 .
Identifying the domain and range The domain of a quadratic function is all real numbers, which can be written as ( − ∞ , ∞ ) . Since the function has a maximum value of 74, the range is all real numbers less than or equal to 74, which can be written as ( − ∞ , 74 ] .
Final Answer Therefore, the function has a maximum value of 74, which occurs at x = 5 . The domain of the function is ( − ∞ , ∞ ) , and the range is ( − ∞ , 74 ] .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, consider a scenario where a company wants to maximize its profit by optimizing the price of a product. The profit function can often be modeled as a quadratic equation, where the x-axis represents the price and the y-axis represents the profit. By finding the vertex of the quadratic function, the company can determine the price that yields the maximum profit. Similarly, in physics, the trajectory of a projectile (like a ball thrown in the air) can be modeled using a quadratic function, allowing us to determine the maximum height the projectile reaches and the time it takes to reach that height. These applications highlight the practical significance of understanding quadratic functions and their properties.
The function f ( x ) = − 3 x 2 + 30 x − 1 has a maximum value of 74, occurring at x = 5 . Its domain is all real numbers ( − ∞ , ∞ ) , and its range is all numbers less than or equal to 74 ( − ∞ , 74 ] .
;
