The function [tex]f(x)=-x^2-4 x+5[/tex] is shown on the graph.
Which statement about the function is true?
A. The domain of the function is all real numbers less than or equal to - 2
B. The domain of the function is all real numbers less than or equal to 9.
C. The range of the function is all real numbers less than or equal to -2
D. The range of the function is all real numbers less than or equal to 9.
Answer (2)
The domain of the function is all real numbers.
Calculate the x-coordinate of the vertex: x = − 2 a b = − 2 .
Calculate the y-coordinate of the vertex: f ( − 2 ) = 9 .
The range of the function is all real numbers less than or equal to 9, so the answer is T h er an g eo f t h e f u n c t i o ni s a ll re a l n u mb ers l ess t han ore q u a lt o 9 .
Explanation
Analyzing the Problem We are given the function f ( x ) = − x 2 − 4 x + 5 and asked to determine which statement about the function is true. The options relate to the domain and range of the function. Since f ( x ) is a quadratic function, its domain is all real numbers. Therefore, we only need to find the range of the function.
Finding the Vertex To find the range, we need to determine the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b , where a = − 1 and b = − 4 . Thus, x = − 2 ( − 1 ) − 4 = − − 2 − 4 = − 2 .
Calculating the Maximum Value Now, we substitute x = − 2 into the function to find the y-coordinate of the vertex, which represents the maximum value of the function: f ( − 2 ) = − ( − 2 ) 2 − 4 ( − 2 ) + 5 = − 4 + 8 + 5 = 9 .
Determining the Range Since the coefficient of the x 2 term is negative ( a = − 1 ), the parabola opens downwards. This means that the vertex represents the maximum point of the function. Therefore, the range of the function is all real numbers less than or equal to 9.
Final Answer Comparing our result with the given options, we find that the correct statement is: The range of the function is all real numbers less than or equal to 9.
Examples
Understanding the range of a quadratic function is useful in various real-world scenarios. For example, if you are modeling the height of a projectile (like a ball thrown in the air) with a quadratic function, the range tells you the maximum height the projectile will reach. Similarly, in business, if you are modeling profit with a quadratic function, the range can help you determine the maximum possible profit.
The function f ( x ) = − x 2 − 4 x + 5 has a domain of all real numbers. The range is all real numbers less than or equal to 9. Therefore, the correct answer is option D: The range of the function is all real numbers less than or equal to 9.
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