Question

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

Answer 1

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]}

Answer 2

`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"`