Y=-x^2-6x+16.
X-intercepts?
Axis of symmetry?
Vertex?
Factored form?
Maximum or minimum?
To find the X-intercepts, we set y = 0 and solve for x:
0 = -x^2 - 6x + 16
x^2 + 6x - 16 = 0
Now, we can use the quadratic formula to find the x-intercepts:
x = (-6 ± sqrt(6^2 - 4*1*(-16)) / (2*1)
x = (-6 ± sqrt(36 + 64)) / 2
x = (-6 ± sqrt(100)) / 2
x = (-6 ± 10) / 2
x = (-6 + 10) / 2 or x = (-6 - 10) / 2
x = 4 / 2 or x = -16 / 2
x = 2 or x = -8
Therefore, the x-intercepts are (2,0) and (-8,0).
The axis of symmetry is a vertical line that passes through the vertex of the parabola, and it can be found using the formula x = -b/(2a). In this case, a=-1 and b=-6, so the axis of symmetry is x = -(-6) / (2*(-1)) = 3.
To find the vertex, we substitute x=3 back into the equation:
y = -(3)^2 - 6(3) + 16
y = -9 - 18 + 16
y = -11
So, the vertex is (3, -11).
To find the factored form, we can rewrite the equation in vertex form:
y = -(x+3)^2 + 25
From the factored form, we can see that the vertex represents the maximum point of the parabola.
Therefore, the maximum value of the function is y = 25 at x = -3.