Inequalities and vertex form

Question

Write these in vertex form

y is less than -.5x^2 - x + 1.5

Answer 1

Given y < -.5 x² - x + 1.5, in the standard form of ax² + bx + c

Vertex form would have this structure:.a(x - h)² + k. It is known that h = -b/2a.

Note that this can be derived from knowing how to "foil" (x - h) by itself, = x² - 2xh + h²

while realizing that the standard form can be rewritten as a(x² + (b/a)x) + c. Therefore, (b/a) = -2h, and rearranging this gives h = -2b/a. And the value of k can be determined by plugging into the vertex form equation.

In this problem, clearly a = -0.5, h = -(-1/(2X(-0.5))) = -1. Plugging into vertex form without k gives

-0.5(x + 1)² = -0.5x² - x - 0.5. Comparing with the original iinequality, -0.5 + k = 1.5, so k = 2.

Therefore, the answer will be y < -0.5(x + 1)² + 2. BTW The coordinates of the vertex are at (h,k)= (-1,2).

Answer 2

y<-0.5x^2-x+1.5

y<-0.5x^2-x +1.5

y<-0.5(x^2+2x)

complete the square with the x-terms

y<-0.5(x^2+2x+1) +1.5 so far but not completed yet !!!

you actually added a "-0.5", not a "1", because the 1 is being multiplied by -0.5

to offset this and not change the right side of the inequality you must add "+0.5"

when you do this you get...

y<-0.5(x+1)^2 +1.5+0.5 which becomes...

y<-0.5(x+1)^2+2 which is the answer

checking our work...

y<-0.5(x^2+2x+1)+2

y<-0.5x^2-x-0.5+2

y<-0.5x^2-x+1.5 which is what you started with