To begin, let's convert the given equation
color(white)("XXX")y=x^2+4x-5XXXy=x2+4x−5
into "vertex form": y=m(x-a)^2+by=m(x−a)2+b (with vertex at (a,b)(a,b))
Completing the square:
color(white)("XXX")y=x^2+4xcolor(green)(+4)-5color(green)(-4)XXXy=x2+4x+4−5−4
Simplifying and writing as a squared binomial:
color(white)("XXX")y=(x+2)^2-9XXXy=(x+2)2−9 (** this is the form I will use for the intercepts later)
or
color(white)("XXX")y=1(x-(-2))^2+(-9)XXXy=1(x−(−2))2+(−9)
which is in vertex form with vertex at (-2,-9)(−2,−9)
The y-intercept is the value of yy when x=0x=0
color(white)("XXX")y=(0+2)^2-9=-5XXXy=(0+2)2−9=−5
The x-intercepts are the values of xx when y=0y=0
color(white)("XXX")0=(x+2)^2-9XXX0=(x+2)2−9
color(white)("XXX")rarr (x+2)^2=9XXX→(x+2)2=9
color(white)("XXX")rarr x+2 = +-3XXX→x+2=±3
color(white)("XXX")rarr x= -5 or +1XXX→x=−5or+1
graph{x^2+4x-5 [-12.37, 10.13, -10.305, 0.945]}
