The graph of the function [tex]f(x)=(x+6)(x+2)[/tex] is shown. Which statements describe the graph? Choose three correct answers.
* The function is negative over ( [tex]$-6,-2$[/tex] ).
* The axis of symmetry is [tex]x=-4[/tex].
* The function is increasing over ( [tex]$-\infty,-4$[/tex] ).
* The domain is all real numbers.
Answer (2)
The function is negative between its roots, which are -6 and -2.
The axis of symmetry is the vertical line through the vertex, x = − 4 .
Polynomial functions like this have a domain of all real numbers.
Therefore, the three correct statements are: the function is negative over ( − 6 , − 2 ) , the axis of symmetry is x = − 4 , and the domain is all real numbers. T h e f u n c t i o ni s n e g a t i v eo v er ( − 6 , − 2 ) ; T h e a x i so f sy mm e t ry i s x = − 4 ; T h e d o maini s a ll re a l n u mb ers
Explanation
Problem Analysis We are given the function f ( x ) = ( x + 6 ) ( x + 2 ) and asked to identify three correct statements describing its graph. Let's analyze each statement.
The function is negative over ( − 6 , − 2 ) .
The axis of symmetry is x = − 4 .
The function is increasing over ( − ∞ , − 4 ) .
The domain is all real numbers.
Sign of the Function First, let's determine where the function is negative. The function f ( x ) = ( x + 6 ) ( x + 2 ) has roots at x = − 6 and x = − 2 . Since the parabola opens upwards (the coefficient of x 2 is positive), the function is negative between the roots. Therefore, f ( x ) < 0 for − 6 < x < − 2 . So, the first statement is correct.
Axis of Symmetry Next, let's find the axis of symmetry. The axis of symmetry is the vertical line that passes through the vertex of the parabola. The x-coordinate of the vertex is the midpoint of the roots. The roots are x = − 6 and x = − 2 , so the x-coordinate of the vertex is
2 − 6 + ( − 2 ) = 2 − 8 = − 4. Therefore, the axis of symmetry is x = − 4 . So, the second statement is correct.
Increasing/Decreasing Intervals Now, let's determine where the function is increasing. Since the parabola opens upwards, the function is decreasing to the left of the vertex and increasing to the right of the vertex. The vertex has an x-coordinate of − 4 , so the function is increasing for -4"> x > − 4 , or over the interval ( − 4 , ∞ ) . Therefore, the third statement, 'The function is increasing over ( − ∞ , − 4 ) ', is incorrect. It should be decreasing over this interval.
Domain of the Function Finally, let's determine the domain of the function. Since f ( x ) = ( x + 6 ) ( x + 2 ) is a polynomial, it is defined for all real numbers. Therefore, the domain is all real numbers. So, the fourth statement is correct.
Conclusion The three correct statements are:
The function is negative over ( − 6 , − 2 ) .
The axis of symmetry is x = − 4 .
The domain is all real numbers.
Examples
Understanding the properties of quadratic functions, such as where they are negative, their axis of symmetry, and their domain, is crucial in various real-world applications. For example, when modeling the trajectory of a projectile, knowing where the function is negative helps determine when the projectile is below ground level. The axis of symmetry can help find the maximum height of the projectile, and the domain ensures the model is valid for all relevant time values. These concepts are also used in optimization problems, such as maximizing profit or minimizing cost, where quadratic functions often appear.
The function f ( x ) = ( x + 6 ) ( x + 2 ) is negative over the interval ( − 6 , − 2 ) , has an axis of symmetry at x = − 4 , and its domain is all real numbers. The incorrect statement is that the function is increasing over ( − ∞ , − 4 ) ; it is actually decreasing in that interval.
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