Consider the function [tex]$f(x)=9 x^2-9 x$[/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 (1)
The function has a minimum because the coefficient of x 2 is positive.
The x-coordinate of the vertex is calculated as x = − b / ( 2 a ) = 0.5 .
The minimum value is found by substituting x = 0.5 into the function: f ( 0.5 ) = − 2.25 .
The domain is all real numbers, and the range is [ − 2.25 , ∞ ) . The function has a minim u m value.
Explanation
Problem Analysis The given function is f ( x ) = 9 x 2 − 9 x . We need to determine if it has a minimum or maximum value, find that value and where it occurs, and identify its domain and range.
Determining Minimum or Maximum Since the coefficient of the x 2 term is positive (9 > 0), the parabola opens upwards, indicating that the function has a minimum value.
Finding the x-coordinate of the vertex To find the x-coordinate of the vertex (where the minimum value occurs), we use the formula x = − b / ( 2 a ) , where a = 9 and b = − 9 . Thus, x = − ( − 9 ) / ( 2 ∗ 9 ) = 9/18 = 0.5 .
Finding the minimum value Now, we find the minimum value by plugging the x-coordinate of the vertex ( x = 0.5 ) into the function: f ( 0.5 ) = 9 ( 0.5 ) 2 − 9 ( 0.5 ) = 9 ( 0.25 ) − 4.5 = 2.25 − 4.5 = − 2.25 . So, the minimum value is -2.25, and it occurs at x = 0.5 .
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 minimum value of -2.25, the range is [ − 2.25 , ∞ ) .
Final Answer The function has a minimum value of -2.25, which occurs at x = 0.5 . The domain of the function is all real numbers, and the range is [ − 2.25 , ∞ ) .
Examples
Understanding the minimum value of a quadratic function is useful in various real-world scenarios. For example, if a company's profit is modeled by a quadratic function, finding the minimum value can help determine the point at which the company starts making a profit after incurring losses. Similarly, in physics, the trajectory of a projectile can be modeled by a quadratic function, and finding the maximum height helps in understanding the projectile's motion. This concept is also applicable in optimization problems where we aim to minimize costs or maximize efficiency.
