Question 414929
Find an equation of the parabola which contains a minimum point
at (2,-1) and ordered pair (-1,7)
<pre><font face = "consolas" color = "indigo" size = 4><b>
Here are the points plotted:

{{{drawing(400,400,-5,15,-8,12, circle(-1,7,.12), circle(2,-1,.12),
locate(-3.6,7,"(-1,7)"), locate(1.8,-1.3,"(2,-1)"),
graph(400,400,-5,15,-8,12) )}}}

The equation of a parabola with vertex (h,k) is

y = a(x - h)² + k

The vertex is the minimum point, so

(h,k) = (2,-1)

We substitute h and k

y = a(x - 2)² + (-1)

y = a(x - 2)² - 1

So we have everything but 'a'.

So we substitute the point (x,y) = (-1,7)

7 = a(-1 - 2)² - 1

 7 = a(-3)² - 1

 7 = a(9) - 1

 7 = 9a - 1

 8 = 9a

{{{8/9}}} = a

So the equation of the parabola is

y = {{{8/9}}}(x - 2)² - 1

{{{drawing(400,400,-5,15,-8,12, circle(-1,7,.1), circle(2,-1,.1),
locate(-3.6,7,"(-1,7)"), locate(1.8,-1.3,"(2,-1)"),
graph(400,400,-5,15,-8,12,(8/9)(x-2)^2-1) )}}}
 
Edwin</pre>