determine the vertex, axis of symmetry, x-intercept, and y-intercept for the equation: y=-2x^2+12x-13.
To determine the vertex, we will use the formula for the x-coordinate of the vertex:
x = -b/2a
In this case, a = -2 and b = 12, so
x = -12 / (2*(-2))
x = -12 / (-4)
x = 3
To find the y-coordinate of the vertex, we will substitute x = 3 into the equation:
y = -2(3)^2 + 12(3) - 13
y = -2(9) + 36 - 13
y = -18 + 36 - 13
y = 5
Therefore, 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 x-coordinate of the vertex is 3. So, the equation of the axis of symmetry is x = 3.
To find the x-intercepts, we will set y = 0 and solve for x:
-2x^2 + 12x - 13 = 0
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -2, b = 12, and c = -13.
x = (-(12) ± √((12)^2 - 4(-2)(-13))) / (2(-2))
x = (-12 ± √(144 - 4(-2)(-13))) / (-4)
x = (-12 ± √(144 - 104)) / (-4)
x = (-12 ± √(40)) / (-4)
x = (-12 ± 2√10) / (-4)
x = 3 ± (1/2)√10
Therefore, the x-intercepts are approximately 1.55 and 4.45.
To find the y-intercept, we will set x = 0 and solve for y:
y = -2(0)^2 + 12(0) - 13
y = 0 + 0 - 13
y = -13
Therefore, the y-intercept is -13.
In summary:
Vertex: (3, 5)
Axis of Symmetry: x = 3
X-intercepts: approximately 1.55 and 4.45
Y-intercept: -13