The vertex of a parabola is (−5,2) , and its focus is (−5,6) .What is the standard form of the parabola??

Accepted Solution

Answer:Step-by-step explanation:If you plot the focus and the vertex, you see that the focus is on the same vertical line, just 4 units up.  Since the vertex is below the focus and a parabola ALWAYS wraps around the focus, the parabola is a positive y equals x-squared type.  Depending upon what you call standard form will dictate how your answer "looks", although they are both the same parabola.  There are 2 forms:[tex](x-h)^2=4p(y-k)[/tex] and[tex]ax^2+bx+c=y[/tex]We will work on the first one, then rewrite it into the second one.Our h value is -5, our k value is 2 (from the vertex (h,k)), and p is defined as the distance between the focus and the vertex.  Our p is 4.  Filling all that in:[tex](x+5)^2=16(y-2)[/tex]That's one form.  If we expand the left side of that form we have[tex]x^2+10x+25=16(y-2)[/tex]  Now divide both sides by 16 to get[tex]\frac{1}{16}x^2+\frac{5}{8}x+\frac{25}{16}=y-2[/tex]Now add 2 to both sides in the form 32/16 to get[tex]\frac{1}{16}x^2+\frac{5}{8}x+\frac{57}{16}=y[/tex]They are both the same parabola; pick whichever one fits your needs.