Quadratic equations are either shown as:
#f(x)=ax^2+bx+c# #color(blue)(" Standard Form")#
#f(x)=a(x-h)^2+k# #color(blue)(" Vertex Form")#
In this case, we'll ignore the #"standard form"# due to our equation being in #"vertex form"#
#"Vertex form"# of quadratics is much easier to graph due to there not being a need to solve for the vertex, it's given to us.
#y=1/2(x-8)^2+2#
#1/2= "Horizontal stretch"#
#8=x"-coordinate of vertex"#
#2=y"-coordinate of vertex"#
It's important to remember that the vertex in the equation is #(-h, k)# so since h is negative by default, our #-8# in the equation actually becomes positive. That being said:
#Vertex = color(red)((8, 2)#
Intercepts are also very easy to calculate:
#y"-intercept:"#
#y=1/2(0-8)^2+2# #color(blue)(" Set " x=0 " in the equation and solve")#
#y=1/2(-8)^2+2# #color(blue)(" "0-8=-8)#
#y=1/2(64)+2# #color(blue)(" "(-8)^2=64)#
#y=32+2# #color(blue)(" "1/2*64/1=64/2=32)#
#y=34# #color(blue)(" "32+2=4)#
#y"-intercept:"# #color(red)((0, 34)#
#x"-intercept:"#
#0=1/2(x-8)^2+2# #color(blue)(" Set " y=0 " in the equation and solve")#
#-2=1/2(x-8)^2# #color(blue)(" Subtract 2 from both sides")#
#-4=(x-8)^2# #color(blue)(" Divide both sides by " 1/2)#
#sqrt(-4)=sqrt((x-8)^2)# #color(blue)(" Square-rooting both removes the square")#
#x"-intercept:"# #color(red)("No Solution")# #color(blue)(" Can't square root negative numbers")#
You can see this to be true, as there are no #x"-intercepts:"#
)
