Given
color(white)("XXX")y=2x^2+8x-4
Our initial objective will be to convert this into "vertex-form",
namely y=color(green)m(x-color(red)a)^2+color(blue)b with vertex at (color(red)a,color(blue)b)
Setting aside the constant (-4) for a moment and extracting the color(green)m factor:
color(white)("XXX")y=color(green)2(x^2+4x)color(white)("XXX")-4
Completing the square:
color(white)("XXX")y=color(green)2(x^2+4xcolor(magenta)(+4))color(white)("XX")-4color(magenta)(-(color(green)2xx4))
Rewriting as a squared binomial and a simplified constant:
color(white)("XXX")y=color(green)2(x+2)^2color(white)("XX")-12
Express in explicit vertex form:
color(white)("XXX")y=color(green)2(x-color(red)(""(-2)))^2+color(blue)(""(-12))
color(orange)("Vertex is at: "(color(red)(-2),color(blue)(-12))
The y-intercept is the value of y when x=0.
Substituting 0 for x in the original equation :
color(white)("XXX")y=2 * 0^2 - 8 * 0 -4
gives the color(cyan)("y-intercept at "y=-4 or, if you prefer at color(cyan)(""(0,-4))
graph{2x^2+8x-4 [-12.78, 12.53, -12.6, 0.06]}
