What characteristics are easily found when given a quadratic equation in Standard form and/or vertex form?

Explain exactly how you determine the characteristics. i. e. In vertex form (h, k) is the vertex of the parabola.

How do you change a quadratic equation in vertex form to standard form? ex: y = 2 (x + 3) ^2 - 1

How do you change a quadratic equation in standard form to vertex form? ex: y = x^2 + 16x + 2

Step-by-step explanation:

a)

y=2 (x + 3) ^2-1

this equation is in vertex form,

it is:

vertically stretched by a factor of 2

left 3 units (f (x) = a * (x - h) ^ (2) + k, so if h is positive it was - (-h) before ot got simplified)

down 1 unit

thus, vertex is (-3,-1)

b)

you can change it to standard form (f (x) = ax^ (2) + bx+c) by:

simply multiplying everything out:

y=2 (x + 3) ^2-1

y=2 (x^ (2) + 6x+9) - 1

y=2x^ (2) + 12x+18-1

y=2x^ (2) + 12x+17

c)

y=x^2+16+2

this equation is in standard form,

you can change it to vertex form (f (x) = a * (x - h) ^ (2) + k by:

y=1 (x-h) ^2+k

h=-b/2a

h=-16/2 (1)

h=-8

solve for k

y=x^2+16+2

since vertex is (h, k) lets plug in h for x to find k and just solve for y:

k=y = (-8) ^2+16+2

k=y=64+16+2

k=y=82

k=82

now that we know the vertex, lets write the equation in vertex form:

y=a * (x - h) ^ (2) + k

y=1 (x - (-8)) ^ (2) + 82

y = (x+8) ^2+82