Question

the standard form of the equation of a parabola is y = x2 - 6x 14. what is the vertex form of the equation?

Answer 1

  y = x^2 - 6x+14
First, group:
y=(x^2-6x)+14
y=(x^2-6x+....)+14-....

Second, add placeholders:

Notice that the second blank is subtracted to account for what you're adding to complete the square. 

Take the coefficient of the x term: -6 
Divide it by 2: -6 / 2 = -3 
Square it: (-3)² = 9 

Add 9 to both blanks.
y=(x^2-6x+9)+14-9
x² - 6x + 9 is the expanded form of a perfect square binomial. 

Remember that (a - b)² = a² - 2ab + b². Apply this to what you have. 

y=(x^2-6x+9)+14-9
y=(x-3)^2+14-9

Now, simplify:
y=(x-30^2+5

Answer 2

The equation of a parabola in standard form can be either of the form $y = ax^2 + bx + c$ (up/down) or of the form $x = ay^2 + by + c$ (left/right).

And we have the up/down standard form form, i.e.,  $y = x^2 - 6x + 14$  (I'm assuming + sign of 14)

Compare the equation of the parabola with the standard form $y = ax^2 + bx + c$ , after comparing, we get, a = 1, b = -6 & c = 14

Find the x-coordinate of the vertex using the formula, $h = \frac{-b}{2a}$

then, we get h = -(-6)/2*1 = 6/2 = 3

to find the y-coordinate (k) of the vertex, substitute x = h in the expression $ax^2+ bx + c$

then k  =  1*3^2 + (-6)3 + 14   =  9 - 18 + 14   =  5

The vertex (h, k) = (3, 5) as an ordered pair

Therefore, the vertex form of the equation is

$y = (x - 3)^2 + 5$

What is the vertex form of quadratic equation?

The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where a, h, and k are constants of the parabola is at (h, k). When the quadratic parent function $f(x) = x^2$ is written in vertex form, $y = a(x - h)^2 + k$, a = 1, h = 0, and k = 0.

What are the 3 form of equation of parabola

Left/Right Opened Parabolas:

Standard form: $x = ay^2 + by + c$

Vertex Form: $x = a (y - k)^2 + h$

Intercept Form: $x = a(y - p)(y - q)$

Top/Bottom Opened Parabolas:

Standard form: $y = ax^2 + bx + c$

Vertex Form: $y = a (x - h)^2 + k$

Intercept Form: $y = a(x - p)(x - q)$

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