Question 478266
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Hi,

Write each equation in standard form. State whether the graph of the equation is a parabola, circle,ellipse, or hyperbola.
 y = x^2 + 3x + 1 
y = (x+3/2)^2 -9/4 + 1 = 
y = (x+3/2)^2 -5/4  |Parabola opening upward with Vertex (-3/2,-5/4)
standard form is {{{(x -h)^2 = 4p(y -k)}}}, (x-3)^2 = y + 5/4

x^2+4y^2+2x-24y+33=0
 (x+1)^2 - 1 + 4(y-3)^2 - 36 + 33 = 0
  (x+1)^2 + 4(y-3)^2 = 4
  (x+1)^2/4 + (y-3)^2 = 1 |Ellipse, C(-1,3) major axis along y = 3
{{{drawing(300,300,   -6, 6, -6, 6,  arc(-1,3,4,2), grid(1),
circle(-1, 3,0.2),
circle(-1.5, -5/4,0.3),
graph( 300, 300, -6, 6, -6, 6,0, (x+3/2)^2 -5/4))}}}

Standard Form of an Equation of a Circle is {{{(x-h)^2 + (y-k)^2 = r^2}}} 
where Pt(h,k) is the center and r is the radius

 Standard Form of an Equation of an Ellipse is {{{(x-h)^2/a^2 + (y-k)^2/b^2 = 1 }}}
where Pt(h,k) is the center and a and b  are the respective vertices distances from center.

Standard Form of an Equation of an Hyperbola opening right and  left is:
  {{{(x-h)^2/a^2 - (y-k)^2/b^2 = 1}}} where Pt(h,k) is a center  with vertices 'a' units right and left of center.
Standard Form of an Equation of an Hyperbola opening up and down is:
  {{{(y-k)^2/b^2 - (x-h)^2/a^2 = 1}}} where Pt(h,k) is a center  with vertices 'b' units up and down from center.

the vertex form of a parabola opening up or down, {{{y=a(x-h)^2 +k}}} where(h,k) is the vertex.The standard form is {{{(x -h)^2 = 4p(y -k)}}}, where  the focus is (h,k + p)
the vertex form of a parabola opening right or left, {{{x=a(y-k)^2 +h}}} where(h,k) is the vertex.The standard form is {{{(y -k)^2 = 4p(x -h)}}}, where  the focus is (h +p,k )