What is the range of the function $f(x)=-(x+3)^2+7$?
A. all real numbers less than or equal to 7
B. all real numbers greater than or equal to 7
C. all real numbers less than or equal to -3
D. all real numbers greater than or equal to -3
Answer (2)
Identify the function as a quadratic function in vertex form.
Determine the vertex of the parabola, which is ( − 3 , 7 ) .
Recognize that the parabola opens downward because the coefficient of the x 2 term is negative.
Conclude that the range of the function is all real numbers less than or equal to 7 , which is expressed as all real numbers less than or equal to 7 .
Explanation
Analyzing the Function The given function is f ( x ) = − ( x + 3 ) 2 + 7 . This is a quadratic function in vertex form. Let's analyze the components to determine the range.
Identifying the Vertex The function is in the form f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, a = − 1 , h = − 3 , and k = 7 . Since a = − 1 is negative, the parabola opens downward. This means the vertex represents the maximum point of the function.
Determining the Range The vertex of the parabola is at ( − 3 , 7 ) . Since the parabola opens downward, the maximum value of the function is 7 . The range consists of all real numbers less than or equal to this maximum value.
Final Answer Therefore, the range of the function f ( x ) = − ( x + 3 ) 2 + 7 is all real numbers less than or equal to 7 .
Examples
Understanding the range of a quadratic function is useful in many real-world scenarios. For example, if you are modeling the height of a ball thrown into the air as a function of time, the range of the function will tell you the maximum height the ball reaches. Similarly, if you are modeling the profit of a business as a function of the number of items sold, the range of the function will tell you the maximum profit the business can achieve. In this case, the function f ( x ) = − ( x + 3 ) 2 + 7 could represent the profit of a company, where the maximum profit is 7 (in some currency unit), and x represents some factor affecting the profit.
The range of the function f ( x ) = − ( x + 3 ) 2 + 7 is all real numbers less than or equal to 7. This is because the vertex of the parabola is at the maximum point (–3, 7) and the parabola opens downwards. Therefore, the correct answer is option A.
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