Graph the function.
$g(x)=-x^2-6 x-6$

Convert the quadratic function to vertex form: g ( x ) = − ( x + 3 ) 2 + 3 .
Identify the vertex: ( − 3 , 3 ) .
Find the y-intercept: ( 0 , − 6 ) .
Find the x-intercepts: ( − 3 + 3 ​ , 0 ) and ( − 3 − 3 ​ , 0 ) .

The graph is a downward-opening parabola with vertex ( − 3 , 3 ) , y-intercept ( 0 , − 6 ) , and x-intercepts ( − 3 ± 3 ​ , 0 ) . g ( x ) = − x 2 − 6 x − 6 ​
Explanation

Analyzing the Quadratic Function We are given the quadratic function g ( x ) = − x 2 − 6 x − 6 . Our goal is to analyze this function and prepare it for graphing. This involves finding key features such as the vertex, intercepts, and the direction in which the parabola opens.

Converting to Vertex Form To find the vertex, we will convert the given quadratic function into vertex form, which is g ( x ) = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola. We start by completing the square for the given function:


g ( x ) = − x 2 − 6 x − 6 g ( x ) = − ( x 2 + 6 x ) − 6
To complete the square, we take half of the coefficient of the x term (which is 6), square it (which gives us 9), and add and subtract it inside the parentheses:
g ( x ) = − ( x 2 + 6 x + 9 − 9 ) − 6 g ( x ) = − (( x + 3 ) 2 − 9 ) − 6
Now, distribute the negative sign:
g ( x ) = − ( x + 3 ) 2 + 9 − 6 g ( x ) = − ( x + 3 ) 2 + 3
So, the vertex form of the given quadratic function is g ( x ) = − ( x + 3 ) 2 + 3 .

Identifying the Vertex From the vertex form g ( x ) = − ( x + 3 ) 2 + 3 , we can identify the vertex of the parabola. The vertex is at the point ( − 3 , 3 ) . Since the coefficient of the ( x + 3 ) 2 term is negative ( − 1 ), the parabola opens downward.

Finding the Y-Intercept To find the y-intercept, we set x = 0 in the original equation:


g ( 0 ) = − ( 0 ) 2 − 6 ( 0 ) − 6 g ( 0 ) = − 6
So, the y-intercept is at the point ( 0 , − 6 ) .

Finding the X-Intercepts To find the x-intercepts, we set g ( x ) = 0 and solve for x :

0 = − x 2 − 6 x − 6 x 2 + 6 x + 6 = 0
We can use the quadratic formula to solve for x :
x = 2 a − b ± b 2 − 4 a c ​ ​
In this case, a = 1 , b = 6 , and c = 6 . Plugging these values into the quadratic formula, we get:
x = 2 ( 1 ) − 6 ± 6 2 − 4 ( 1 ) ( 6 ) ​ ​ x = 2 − 6 ± 36 − 24 ​ ​ x = 2 − 6 ± 12 ​ ​ x = 2 − 6 ± 2 3 ​ ​ x = − 3 ± 3 ​
So, the x-intercepts are x = − 3 + 3 ​ ≈ − 1.268 and x = − 3 − 3 ​ ≈ − 4.732 . The x-intercepts are approximately ( − 1.268 , 0 ) and ( − 4.732 , 0 ) .

Graphing the Parabola In summary, we have found the following key features of the quadratic function g ( x ) = − x 2 − 6 x − 6 :


Vertex: ( − 3 , 3 )
Y-intercept: ( 0 , − 6 )
X-intercepts: ( − 3 + 3 ​ , 0 ) ≈ ( − 1.268 , 0 ) and ( − 3 − 3 ​ , 0 ) ≈ ( − 4.732 , 0 )

These features allow us to accurately graph the parabola. The parabola opens downwards from the vertex ( − 3 , 3 ) , passes through the y-intercept ( 0 , − 6 ) , and crosses the x-axis at approximately ( − 1.268 , 0 ) and ( − 4.732 , 0 ) .
Examples
Understanding 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 initial velocity and launch angle, they can predict the range and maximum height of the projectile. Similarly, architects use quadratic functions to design arches and suspension bridges, ensuring structural stability and optimal load distribution. The vertex of the parabola helps determine the maximum or minimum value in these scenarios, providing critical information for design and safety.

Answered by GinnyAnswer | 2025-07-08