What is the equation of the parabola that has a vertex at # (5, 4) # and passes through point # (7,-8) #?

2 Answers
Binayaka C.
Oct 9, 2017

The equation of parabola is # y = -3x^2+30x-71 #

Explanation:

The equation of parabola in vertex form is #y= a(x-h)^2+k#

# (h,k)# being vertex here #h=5 , k=4 :. # Equation of parabola in

vertex form is #y= a(x-5)^2+4# . The parabola passes through

point #(7 ,-8)# . So the point #(7 ,-8)# will satisfy the equation .

# :. -8 = a( 7-5)^2 +4 or -8 = 4a +4 # or

#4a = -8-4 or a = -12/4= -3 # Hence the equation of

parabola is #y= -3(x-5)^2+4# or

# y= -3(x^2-10x+25)+4 or y = -3x^2+30x-75+4 # or

#y = -3x^2+30x-71 #

graph{-3x^2+30x-71 [-20, 20, -10, 10]}

Jim G.
Oct 9, 2017

#y=-3x^2+30x-71#

Explanation:

#"the equation of a parabola in "color(blue)"vertex form"# is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))#

#"where "(h,k)" are the coordinates of the vertex and a"#
#"is a multiplier"#

#"here "(h,k)=(5,4)#

#rArry=a(x-5)^2+4#

#"to find a substitute "(7,-8)" into the equation"#

#-8=4a+4rArra=-3#

#rArry=-3(x-5)^2+4larrcolor(red)" in vertex form"#

#"distributing and simplifying gives"#

#y=-3(x^2-10x+25)+4#

#color(white)(y)=-3x^2+30x-75+4#

#rArry=-3x^2+30x-71larrcolor(red)" in standard form"#

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