Determine the vertex,axis symmetry, x-intercept, and y-intercept for the equation: y = -2x^2 + 12x - 13
To determine the vertex of the given equation, we need to find the axis of symmetry. The equation of the axis of symmetry is given by x = -b/(2a), where the equation is in the form y = ax^2 + bx + c.
In this case, a = -2 and b = 12. Plugging these values into the equation, we have:
x = -12/(2*(-2))
x = -12/(-4)
x = 3
So, the axis of symmetry is x = 3. To find the vertex, we substitute this value back into the equation:
y = -2(3)^2 + 12(3) - 13
y = -18 + 36 - 13
y = 5
Therefore, the vertex of the equation is (3, 5).
The axis of symmetry is x = 3.
To find the x-intercepts, we set y = 0:
0 = -2x^2 + 12x - 13
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, factoring is not possible, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = -2, b = 12, and c = -13, 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 ± (1/2)√10
So, the x-intercepts are x = 3 + (1/2)√10 and x = 3 - (1/2)√10.
To find the y-intercept, we set x = 0:
y = -2(0)^2 + 12(0) - 13
y = -13
Therefore, the y-intercept is -13.
In summary, the vertex is (3, 5), the axis of symmetry is x = 3, the x-intercepts are x = 3 + (1/2)√10 and x = 3 - (1/2)√10, and the y-intercept is -13.