To graph the equation y = -x^2 - 10x - 16, we can plot several points to create the graph. Let's find the vertex and the root first, and then plot the additional points.
1. Vertex:
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a and b are the coefficients of x^2 and x, respectively. In this equation, a = -1 and b = -10.
x = -(-10) / (2 * -1) = 10 / -2 = -5
To find the corresponding y-coordinate, substitute the x-coordinate into the equation:
y = -(-5)^2 - 10(-5) - 16
y = -25 + 50 - 16
y = 9
So, the vertex is (-5, 9).
2. Root:
To find the root, set y = 0 and solve for x:
0 = -x^2 - 10x - 16
This equation does not factor easily, so we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this equation, a = -1, b = -10, and c = -16.
x = (-(-10) ± √((-10)^2 - 4(-1)(-16))) / (2 * -1)
x = (10 ± √(100 - 64)) / -2
x = (10 ± √36) / -2
x = (10 ± 6) / -2
The two possible roots are:
x1 = (10 + 6) / -2 = 16 / -2 = -8
x2 = (10 - 6) / -2 = 4 / -2 = -2
So, the roots are -8 and -2.
Now, let's plot these points and two additional points:
Point 1: Vertex (-5, 9)
Point 2: Root (-8, 0)
Point 3: Root (-2, 0)
Point 4: x = -6, y = -10(-6)^2 - 10(-6) - 16 = -196
Point 5: x = 0, y = -16
Here's the graph of the equation y = -x^2 - 10x - 16:
```
|
10 +
| *
8 + *
|
6 +
|
4 +
|
2 +
| *
0 + * * * * * * * * * * * * *
-------------------------------------------------
-8 -6 -4 -2 0 2 4 6 8 10 12 14 16
```
The graph is a downward-opening parabola that intersects the x-axis at -8 and -2 (roots) and reaches a maximum point at the vertex (-5, 9).