Determine the vertex, axis of symmetry, x-intercept, and y -intercept for the equation: y=−2x2+12x−13 .
To determine the vertex of the equation, we need to find the x-coordinate of the vertex using the formula x = -b/2a.
In this case, a = -2 and b = 12.
x = -12 / (2*(-2))
x = -12 / (-4)
x = 3
To find the y-coordinate of the vertex, we substitute the x-value into the equation:
y = -2*(3)^2 + 12*3 - 13
y = -2*9 + 36 - 13
y = -18 + 36 - 13
y = 5
So, the vertex of the equation is (3, 5).
The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is x = 3.
To find the x-intercepts, we set the equation equal to zero and solve for x:
-2x^2 + 12x - 13 = 0
Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), we have:
x = (-12 ± √(12^2 - 4*(-2)*(-13))) / (2*(-2))
x = (-12 ± √(144 - 104)) / (-4)
x = (-12 ± √(40)) / (-4)
x = (-12 ± 2√10) / (-4)
x = (3 ± √10) / 2
So, the x-intercepts are (3 + √10)/2 and (3 - √10)/2.
To find the y-intercept, we substitute x = 0 into the equation:
y = -2*(0)^2 + 12*(0) - 13
y = -13
So, the y-intercept is (0, -13).