Graph the equation y=-x^2+12x-32 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.
To graph the equation y = -x^2 + 12x - 32, we need to find the roots and the vertex first.
The equation is in the standard form of a quadratic equation, y = ax^2 + bx + c, where a = -1, b = 12, and c = -32.
To find the roots, we set y = 0 and solve for x:
0 = -x^2 + 12x - 32
Rearranging the equation:
x^2 - 12x + 32 = 0
Factoring the quadratic equation:
(x - 4)(x - 8) = 0
Setting each factor equal to zero:
x - 4 = 0 or x - 8 = 0
x = 4 x = 8
So, the roots of the equation are x = 4 and x = 8.
Next, to find the vertex, we use the formula x = -b/2a, where a = -1 and b = 12:
x = -12 / (2*(-1))
x = -12 / (-2)
x = 6
Now we substitute x = 6 back into the equation to find the y-coordinate of the vertex:
y = -(6)^2 + 12(6) - 32
y = -36 + 72 - 32
y = 4
So, the vertex is (6, 4).
Now we can plot the points: roots (4, 0) and (8, 0), and vertex (6, 4).
Here is the graph:
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- 4 - 2 - 0 - 2 - 4 - 6 - 8 - 10 - 12