Graph the equation y=-x^2-8x-7on the accompanying set of axes. You must plot 5 points including the roots and the vertex.
To graph the equation y = -x^2 - 8x - 7, we can start by finding the x-intercepts (roots) and the vertex.
1. Finding the x-intercepts (roots):
To find the x-intercepts, we set y = 0 and solve for x.
0 = -x^2 - 8x - 7
Rearranging the equation: x^2 + 8x + 7 = 0
Factoring the equation: (x + 7)(x + 1) = 0
Setting each factor equal to zero:
x + 7 = 0 => x = -7
x + 1 = 0 => x = -1
So the x-intercepts (roots) are x = -7 and x = -1.
2. Finding the vertex:
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -1 and b = -8.
x = -(-8) / 2(-1)
x = 8 / 2
x = 4
To find the y-coordinate of the vertex, we substitute x = 4 into the equation: y = -x^2 - 8x - 7.
y = -(4)^2 - 8(4) - 7
y = -16 - 32 - 7
y = -55
So the vertex is (4, -55).
Now, let's plot these points on the graph:
- x-intercepts (roots): (-7, 0) and (-1, 0)
- vertex: (4, -55)
Here's the graph of y = -x^2 - 8x - 7 with the points plotted:
{{{ graph( 400, 400, -10, 10, -60, 10, -x^2 - 8x - 7) }}}