The graph of the function [tex]f(x)=x^2+8 x+12[/tex] is shown. Which statements describe the graph? Check all that apply.

A. The vertex is the maximum value.
B. The axis of symmetry is [tex]x=-4[/tex].
C. The domain is all real numbers.
D. The range is all real numbers.
E. The function is increasing over ([tex]-\infty,-4[/tex]).
F. The [tex]x[/tex]-intercepts are at [tex](-6,0)[/tex] and [tex](-2,0)[/tex].

The true statements about the graph of the function f ( x ) = x 2 + 8 x + 12 are: the axis of symmetry is x = − 4 , the domain is all real numbers, and the x-intercepts are at ( − 6 , 0 ) and ( − 2 , 0 ) . The statements A, D, and E are false.
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Answered by Anonymous | 2025-07-05

The vertex is a minimum, not a maximum, because the parabola opens upwards.
The axis of symmetry is found using x = − b /2 a , which is x = − 4 .
The domain of a quadratic function is all real numbers.
The x-intercepts are found by solving x 2 + 8 x + 12 = 0 , which gives x = − 6 and x = − 2 . Thus, the final answer is: The axis of symmetry is x = − 4 , the domain is all real numbers, and the x -intercepts are at ( − 6 , 0 ) and ( − 2 , 0 ) . x = − 4 , domain: all real numbers , ( − 6 , 0 ) , ( − 2 , 0 ) ​

Explanation

Analyzing the Statements We are given the function f ( x ) = x 2 + 8 x + 12 and asked to determine which statements accurately describe its graph. Let's analyze each statement.

The vertex is the maximum value. Since the coefficient of the x 2 term is positive (1), the parabola opens upwards. This means the vertex represents the minimum value of the function, not the maximum. Therefore, this statement is false.

The axis of symmetry is x = − 4 . The axis of symmetry for a quadratic function in the form f ( x ) = a x 2 + b x + c is given by the formula x = − 2 a b ​ . In this case, a = 1 and b = 8 , so the axis of symmetry is x = − 2 ( 1 ) 8 ​ = − 4 . This statement is true.

The domain is all real numbers. Since f ( x ) is a polynomial, it is defined for all real numbers. Therefore, the domain is all real numbers. This statement is true.

The range is all real numbers. Since the parabola opens upwards, the function has a minimum value at the vertex. To find the y-coordinate of the vertex, we can plug x = − 4 into the function: f ( − 4 ) = ( − 4 ) 2 + 8 ( − 4 ) + 12 = 16 − 32 + 12 = − 4 . Thus, the vertex is at ( − 4 , − 4 ) . Since the parabola opens upwards, the range is all real numbers greater than or equal to -4, i.e., y ≥ − 4 . Therefore, this statement is false.

The function is increasing over ( − ∞ , − 4 ) . Since the parabola opens upwards and the vertex is at x = − 4 , the function is decreasing to the left of the vertex and increasing to the right of the vertex. Therefore, the function is decreasing over ( − ∞ , − 4 ) and increasing over ( − 4 , ∞ ) . This statement is false.

The x -intercepts are at ( − 6 , 0 ) and ( − 2 , 0 ) . To find the x-intercepts, we set f ( x ) = 0 and solve for x : x 2 + 8 x + 12 = 0 . We can factor this quadratic as ( x + 6 ) ( x + 2 ) = 0 . Therefore, the x-intercepts are at x = − 6 and x = − 2 . This statement is true.

Calculations and Key Points The vertex of the parabola is at x = − 2 a b ​ = − 2 ( 1 ) 8 ​ = − 4 . The y-coordinate of the vertex is f ( − 4 ) = ( − 4 ) 2 + 8 ( − 4 ) + 12 = 16 − 32 + 12 = − 4 . So the vertex is ( − 4 , − 4 ) .


The axis of symmetry is x = − 4 .
The domain of the quadratic function is all real numbers.
The range of the quadratic function is y ≥ − 4 .
The function is decreasing on ( − ∞ , − 4 ) and increasing on ( − 4 , ∞ ) .
To find the x-intercepts, we solve x 2 + 8 x + 12 = 0 . Factoring gives ( x + 6 ) ( x + 2 ) = 0 , so x = − 6 or x = − 2 . The x-intercepts are ( − 6 , 0 ) and ( − 2 , 0 ) .

Identifying Correct Statements Based on our analysis, the correct statements are:


The axis of symmetry is x = − 4 .
The domain is all real numbers.
The x -intercepts are at ( − 6 , 0 ) and ( − 2 , 0 ) .


Final Answer The correct statements describing the graph of the function f ( x ) = x 2 + 8 x + 12 are:


The axis of symmetry is x = − 4 .
The domain is all real numbers.
The x -intercepts are at ( − 6 , 0 ) and ( − 2 , 0 ) .

Examples
Understanding the properties of quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By knowing the vertex, axis of symmetry, domain, range, and intercepts of the quadratic function, they can accurately predict the maximum height, landing point, and overall path of the projectile. This knowledge is essential for designing efficient and safe systems in fields like aerospace, sports, and construction. For example, if you want to throw a ball as far as possible, understanding the quadratic trajectory helps you determine the optimal angle and initial velocity to achieve maximum range. The x-intercepts would represent where the ball lands on the ground.

Answered by GinnyAnswer | 2025-07-05