y=-x^2+8x-5?

Question

Find the axis of symmetry. Please show work. I do not understand how to find the symmetry.

Answer 1

I think it's best to just use completing the square to get the equation in the form of (y - k) = 4p(x - h)^2, because then you know that the vertex point is at (h, k) and so the axis of symmetry must be the vertical line that passes through this, which is x=h.

y + 5 = -x^2 + 8x
y + 5 = -(x^2 - 8x)
y + 5 - 16 = -(x^2 - 8x + 16)
y - 11 = -(x - 4)^2
So the axis of symmetry for this parabola is x = 4.

Similarly, you could just find two values of x that give back the same y, and take the point in between the two x values. This midpoint would have to be along the axis of symmetry, which you can see by drawing a parabola. You CAN'T always expect two use two values where y = 0 (which would just be the roots of ax^2 + bx+ c = 0), because it's possible that the equation has no roots, which means the parabola never touches the x-axis.

Answer 2

in the quadratic function
a = -1 , b = 8 , c = -5

x = -b / 2a

= -8 / (2 x -1) = 4

the axis of symmetry is : x = 4

Answer 3

the equation of axis symmetry for y=ax^2+bx=c
is x=-b/2a
for exam x=-8/2(-1)====> x=4 is axis symm for y=-x^2+8x-5

Answer 4

y = -(x^2 - 8x + 5)

Find the roots

The axis of symmetry lies at the midpoint between the roots

x = (8 +/- SQRT(64 - 4.5))/2

= 4 +/- SQRT(44)/2

So midpoint is (4 + SQRT(44)/2 + 4 - SQRT(44)/2) /2

x = 4

Answer 5

y = x^2 + 8x - 5
y = x^2 + 8x + 16 - 16 - 5
y = (x + 4)^2 - 21
y + 21 = (x + 4)^2
axis of symmetry is x = -4