The function $f(x)=-0.3(x-5)^2+5$ is graphed. What are some of its key features? Check all that apply.
The axis of symmetry is $x=5$.
The domain is ${x \mid x$ is a real number $}$.
The function is increasing over ( $-\infty, 5$ ).
The minimum is $(5,5)$.
The range is ${y \mid y \geq 5}$.
Answer (2)
The function is a quadratic function in vertex form, f ( x ) = − 0.3 ( x − 5 ) 2 + 5 , with vertex at ( 5 , 5 ) .
The axis of symmetry is x = 5 .
The domain is all real numbers, { x ∣ x is a real number } , and the function is increasing on ( − ∞ , 5 ) .
The range is { y ∣ y ≤ 5 } .
Explanation
Identifying the Vertex Form The given function is f ( x ) = − 0.3 ( x − 5 ) 2 + 5 . This is a quadratic function in vertex form, f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In this case, a = − 0.3 , h = 5 , and k = 5 .
Determining the Vertex and Direction The vertex of the parabola is ( 5 , 5 ) . Since a = − 0.3 < 0 , the parabola opens downwards. This means that the vertex represents the maximum point of the function.
Finding the Axis of Symmetry The axis of symmetry is a vertical line that passes through the vertex. Therefore, the axis of symmetry is x = 5 .
Determining the Domain The domain of any quadratic function is all real numbers, since we can input any real number into the function. So, the domain is { x ∣ x is a real number } .
Finding the Increasing Interval Since the parabola opens downwards and the vertex is ( 5 , 5 ) , the function is increasing on the interval ( − ∞ , 5 ) and decreasing on the interval ( 5 , ∞ ) .
Identifying the Maximum and Minimum Since the parabola opens downwards, the vertex ( 5 , 5 ) is the maximum point. Thus, the function has a maximum value of 5 at x = 5 , and there is no minimum.
Determining the Range Since the parabola opens downwards and the maximum value is 5, the range of the function is all real numbers less than or equal to 5. Therefore, the range is { y ∣ y ≤ 5 } .
Checking the Options Now, let's check the given options:
The axis of symmetry is x = 5 . This is correct.
The domain is { x ∣ x is a real number } . This is correct.
The function is increasing over ( − ∞ , 5 ) . This is correct.
The minimum is ( 5 , 5 ) . This is incorrect, as ( 5 , 5 ) is the maximum.
The range is { y ∣ y ≥ 5 } . This is incorrect, the range is { y ∣ y ≤ 5 } .
Examples
Quadratic functions are used in various real-world applications, such as modeling the trajectory of a projectile, designing parabolic reflectors for antennas and solar ovens, and determining the optimal dimensions for maximizing area or minimizing cost. For example, if you throw a ball, its path can be modeled by a quadratic function, where the height of the ball at any given time can be predicted using the equation. Understanding the key features of quadratic functions, like the vertex and axis of symmetry, helps in analyzing and optimizing these real-world scenarios.
The function f ( x ) = − 0.3 ( x − 5 ) 2 + 5 has a vertex at ( 5 , 5 ) and the axis of symmetry is x = 5 . It is increasing on the interval ( − ∞ , 5 ) , and its range is { y ∣ y ≤ 5 } . The function does not have a minimum, as the vertex represents the maximum point.
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