why is the axis of symmetry x= -b/2a?

Question

we are learning about coordinates, vertex, domains, linear equations (if thats helpful).... thanks!!

Answer 1

The axis of symmetry is -b/2a because it is precisely the coordinate of the parabola's vertex.

Recall that a general parabola in vertex form is the following:

y = a(x - h) + k

where (h, k) represents the coordinates of the vertex.

Suppose we have a general parabola,
y = ax^2 + bx + c

If we complete the square, we can put this into vertex form.

y = a(x^2 + (b/a)x) + c

y = a(x^2 + (b/a)x + (b^2)/(4a^2) ) + c - b^2/(4a)

y = a(x + b/[2a])^2 + [4ac - b^2]/[4a]

This means the coordinates of the vertex is

(-b/[2a] , [4ac - b^2]/[4a])

And as you can see, -b/[2a] is the vertex's x-coordinate and subsequently the axis of symmetry.

Answer 2

a million. y = -x² + 6x - a million Use partial factoring which excludes the consistent, the (-a million) out of your calculation. you at present have: g =-x² + 6x Now ingredient the the stunning decision ingredient: g =- x(x - 6) Now set g corresponding to 0 and scientific care utilising the severe of 0 products: g = 0 0 =-x(x-6) x=0 x=6 ultimately, upload those 2 concepts and then divide by ability of two. (0 + 6) / 2 = 3 x = 3 is the situation the line of symmetry lies. 2. y = x² - 2x + 4 enable y = 0 0 = x² -2x + 4 finished the sq.: -4 = (x² -2x) -4 = (x² -2x + a million) - a million -3 = (x - a million )² Take the sq. root of the two factors! yet, taking the sq. root of -3 ought to contain utilising (i) it quite is an imaginary quantity and nevertheless it quite is sq. rooted`, i'm not sure that's what you seek for for. This quadratic would not have roots (x intercepts) indoors the definitely quantity gadget. you have used the gadget referred to as the `discriminant`to make sure from the outset that there have been no roots indoors the definitely quantity worldwide!

Answer 3

The axis of symmetry is found halfway between the roots, making it the average of the roots. By the Quadratic Formula, the roots arex = -b +- sqrt(b^2 - 4ac) all over 2a.

If you average these by adding then dividing by 2, the sqrt parts drop out and you get (-2b over 2a) divided by 2 which is -b over 2a.

Answer 4

since -b/2a is the x-coordinate for the vertex, it is right in the middle of the parabola.

therefore, x = -b/2a is the axis of symmetry for any upward- or downward-opening parabola