Question

What is the vertex of `y = 1/2 (3x + 4)^2 - 5`?

Answer 1

`color(blue)("Vertex"->(x,y)->(-4/3,-5)`

Explanation:

Consider the following:

Standard form`->y=ax^2+bx+c`

Vertex form`->y=a(x+b/(2a))^2+k+c`

Where `k=(-1)xxa(b/(2a))^2`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("The given equation is not quite in vertex form")`

Write as:`" "y=3/2(x+4/3)^2-5" "` Now it is!

`color(blue)(x_("vertex")=color(brown)((-1)xxb/(2a))color(green)( = (-1)xx4/3 )= -4/3`
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`color(blue)(y_("vertex")=color(brown)(k+c) =-5`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Vertex"->(x,y)->(-4/3,-5)`