Question

How do you write the quadratic function in vertex form given vertex (4,5) and point (8,-3)?

Answer 1

`y = -1/2(x-4)^2+5`

Explanation:

Vertex form looks like this:

`y = a(x-h)^2+k`

where `a` is a constant multiplier affecting the "steepness" of the parabola and whether it is upright or inverted and `(h, k)` is the vertex.

So in our example, we are looking for an equation of the form:

`y = a(x-color(blue)(4))^2+color(blue)(5)`

In order that this pass through the point `(8, -3)` we require that these coordinates satisfy the equation, so:

`color(blue)(-3) = a(color(blue)(8)-4)^2+5`

`color(white)(-3) = 16a+5`

Subtract `5` from both ends to get:

`-8 = 16a`

Divide both sides by `16` to find:

`a = -1/2`

So the equation we want is:

`y = -1/2(x-4)^2+5`

graph{(y+1/2(x-4)^2-5)((x-4)^2+(y-5)^2-0.01)((x-8)^2+(y+3)^2-0.01) = 0 [-6.21, 13.79, -4.16, 5.84]}