Question

How do you find the vertex and the intercepts for `y = 1/4x^2 + 1/ 2x - 3/4`?

Answer 1

Vertex is at `(-1,-1)`, x intercepts are at `(-3,0) and (1,0)`
and y intercept is at `(0,-3/4)`,

Explanation:

`y= 1/4 x^2+1/2 x -3/4` . Comparing with standard equation

`y=ax^2+b+c` we get `a= 1/4 , b = 1/2 ,c = -3/4` We know

vertex `(x) = -b/(2a) = -1/2/(2*1/4) = -1`. Putting `x= -1`

in the equation `y= 1/4 x^2+1/2 x -3/4`,

we get, vertex `(y) = 1/4 (-1)^2 +1/2(-1)-3/4 =1/4-1/2-3/4`

`= -4/4= -1` .Vertex is at `(-1,-1)` . Putting `x=0`

in the equation we can get y intercept as `y= -3/4 or (0,-3/4)`,

putting `y=0` in the equation we can get x intercepts as

`0= 1/4 x^2+1/2 x -3/4 or x^2+2x-3=0`

or `(x+3)(x-1)=0 :. x = -3 or x = 1` , x intercepts are at

`(-3,0) and (1,0)`

graph{1/4x^2+1/2x-3/4 [-10, 10, -5, 5]}

Answer 2

Alternative approach
`y_("intercept") ->(x,y)=(0,-3/4)`

Vertex`->(x,y)=(-1,-1)`

`x_("intercepts")-> x=-3 and x=1`

Explanation:

Given: `y=1/4x^2+1/2x-3/4`

Compare to the form of `y=ax^2+bx+c`

`color(blue)("Determine the y intercept")`

`y_("intercept")=c=-3/4 ->(x,y)=(0,-3/4)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Determine the vertex")`

You have three methods:

1. Using the completed square form equation you can read it directly off but with a small amount of adjustment.
   Complete breakdown on this can be found on https://socratic.org/s/aHr8G5h4

2.factorising or use the formula to determine the `x` intercepts and the `x` value of the vertex will be half way between

1. You can do it this way:
   Note that this process is part of that for completing the square.

Write as:

`y=1/4(x^2color(red)(+2x))-3/4`

`x_("vertex") = (-1/2)xx(color(red)(+2)) = -1`

Buy substitution

`y_("vertex")=1/4(-1)^2+1/2(-1)-3/4 = -1`

Vertex`->(x,y)=(-1,-1)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the "x_("interpts")`

Using other method
`y=ax^2+bx+c =0color(white)(..)` where `x=(-b+-sqrt(b^2-4ac))/(2a)`

`a=1/4; b=1/2; c= -3/4`

`x=(-1/2+-sqrt((1/2)^2-4(1/4)(-3/4)))/(2xx1/4)`

`x=-1+-2sqrt( 1/4+3/4)`

`x=-1+-2`

`x=-3 and x=1`