What is the equation of the parabola that has a vertex at # (9, -23) # and passes through point # (35,17) #?

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Answer 1 · 2016-01-13T01:37:00

We can solve this using the vertex formula, #y=a(x-h)^2+k#

The standard format for a parabola is

#y = ax^2 + bx + c#

But there is also the vertex formula,

#y=a(x-h)^2+k#

Where #(h,k)# is the location of the vertex.

So from the question, the equation would be

#y=a(x-9)^2-23#

To find a, substitute the x and y values given: #(35,17)# and solve for #a#:

#17=a(35-9)^2-23#

#(17+23)/(35-9)^2=a#

#a=40/26^2 = 10/169#

so the formula, in vertex form, is

#y = 10/169(x-9)^2-23#

To find the standard form, expand the #(x-9)^2# term, and simplify to
#y = ax^2 + bx + c# form.

Answer 2 · 2016-01-13T01:41:02

For problems of this type, use vertex form, y = a#(x - p)^2# + q.

In vertex form, mentioned above, the vertex's coordinates are (p, q) and a point (x , y) that is on the parabola.

When finding the equation of the parabola, we have to solve for a, which influences the width and the direction of opening of the parabola.

y = a#(x - p)^2# + q
17 = a#(35 - 9)^2# - 23
17 = 576a - 23
17 + 23 = 576a
#5/72# = a

So, the equation of the parabola is y = #5/72##(x - 9)^2# - 23.

Hopefully you understand now!