This is graph of an inverted parabola. To find the vertex (the maximum turning point) write the equation in the form
y = acolor(blue)((x+b)^2) color(magenta)(+c) y=a(x+b)2+c which gives us the vertex as (-b, c)(−b,c)
We use the process of "completing the square"
Note:(x-5)^2 = x^2 -10x+25 " in particular" (10/2)^2 = 25(x−5)2=x2−10x+25 in particular(102)2=25
In the square of a binomial , this relationship is always true.
color(red)((b/2)^2 )= c(b2)2=c
STEP 1 make the coefficient of x^2x2 term = +1.
y = -(1x^2 +3x -12) color(white)(xx)(-12 " is not the correct value for c")y=−(1x2+3x−12)×(−12 is not the correct value for c) .
STEP 2 add and subtract color(red)((b/2)^2 )" which is " (3/2)^2(b2)2 which is (32)2
y = -[color(blue)(x^2 +3x +(3/2)^2) color(magenta)(-(3/2)^2-12])y=−[x2+3x+(32)2−(32)2−12)
STEP 3 Write the first 3 terms as a perfect square
y = -[color(blue)((x+3/2)^2) + color(magenta)((-(9/4)-48/4))]y=−[(x+32)2+(−(94)−484)]
STEP 4 Simplify the constant
y = -[ color(blue)((x+3/2)^2)color(magenta)(-57/4)]y=−[(x+32)2−574]
STEP 5 Remove the outer bracket to get vertex form.
y = - color(blue)((x+3/2)^2)color(magenta)(+57/4)y=−(x+32)2+574
y = acolor(blue)((x+b)^2) color(magenta)(+c) y=a(x+b)2+c
The vertex is at (-3/2, +57/4)(−32,+574)
graph{-x^2 -3x+12 [-3.834, 1.166, 12.52, 15.02]}
