Determine the vertex, axis of symmetry, x-intercept, and y -intercept for the equation: y=−2x2+12x−13%0D%0A%0D%0A=%0D%0A−%0D%0A2%0D%0A%0D%0A2%0D%0A+%0D%0A12%0D%0A%0D%0A−%0D%0A13%0D%0A.
To find the vertex, we can use the formula:
x = -b / (2a)
For the given equation, a = -2 and b = 12. Substituting these values into the equation, we get:
x = -(12) / (2(-2))
x = -12 / -4
x = 3
To find the y-coordinate of the vertex, we can substitute this value of x back into the equation:
y = -2(3)^2 + 12(3) - 13
y = -18 + 36 - 13
y = 5
Therefore, the vertex is (3, 5).
The axis of symmetry is a vertical line passing through the x-coordinate of the vertex. In this case, the equation of the axis of symmetry is:
x = 3
To find the x-intercepts, we need to solve the equation when y is equal to zero. So, we set y to zero and solve for x:
0 = -2x^2 + 12x - 13
This quadratic equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = -2, b = 12, and c = -13. Substituting these values into the quadratic formula, we get:
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
Therefore, the x-intercepts are x = 3 + √10 and x = 3 - √10.
To find the y-intercept, we substitute x = 0 into the equation:
y = -2(0)^2 + 12(0) - 13
y = 0 + 0 - 13
y = -13
Therefore, the y-intercept is (0, -13).