The function [tex]f(x)=-0.3(x-5)^2+5[/tex] is graphed. What are some of its key features? Choose three correct answers.
* The domain is [tex] \{x \mid x[/tex] is a real number [tex]\}[/tex].
* The axis of symmetry is [tex]x=5[/tex].
* The minimum is (5, 5).
* The range is [tex]\{y \mid y \geq 5\}[/tex].
Answer (1)
The domain of the quadratic function f ( x ) = − 0.3 ( x − 5 ) 2 + 5 is all real numbers.
The axis of symmetry is x = 5 .
The function has a maximum at ( 5 , 5 ) , and the range is y ≤ 5 .
Explanation
Analyzing the Function We are given the function f ( x ) = − 0.3 ( x − 5 ) 2 + 5 and asked to identify three correct key features from the given options. Let's analyze each option.
Identifying the Form of the Function The function is a quadratic function in vertex form, f ( x ) = a ( x − h ) 2 + k , where a = − 0.3 , h = 5 , and k = 5 . Since a < 0 , the parabola opens downward, and the vertex represents a maximum point.
Determining the Domain The domain of a quadratic function is all real numbers, so the domain is x ∣ x is a real number . This statement is correct.
Determining the Axis of Symmetry The axis of symmetry for a quadratic function in vertex form is x = h , so the axis of symmetry is x = 5 . This statement is correct.
Determining Minimum or Maximum Since the parabola opens downward, the function has a maximum value at the vertex, not a minimum. The vertex is at ( 5 , 5 ) , so the maximum value is 5. Therefore, the statement "The minimum is (5, 5)" is incorrect.
Determining the Range Since the parabola opens downward and the maximum value is 5, the range is y ∣ y ≤ 5 . The statement "The range is y ∣ y ≥ 5 " is incorrect.
Final Answer Therefore, the three correct answers are:
The domain is x ∣ x is a real number .
The axis of symmetry is x = 5 .
Examples
Understanding the properties of quadratic functions is crucial in various real-world applications. For instance, when designing a bridge, engineers use quadratic functions to model the parabolic shape of the bridge's arch. By analyzing the key features such as the vertex (maximum height) and axis of symmetry, they can ensure the structural integrity and stability of the bridge. Similarly, in projectile motion, the path of a projectile (like a ball thrown in the air) can be modeled using a quadratic function, allowing us to determine its maximum height and range.
