Integrated Math 1
posted by Tifini on .
Suppose a parabola f(x) has its vertex at (0, 25) and its zeros at x = 5 and x = 5. Then f(x)equals _____.
A. 25x^2
B. 1 1/25 x^2
C. x^21
D. x^225

The standard equation of a parabola is :
y = a x ^ 2 + bx + c
The vertex of a parabola is the point where the parabola crosses its axis.
If the coefficient of the x ^ 2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”shape.
If the coefficient of the x ^ 2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “U”shape
But the equation for a parabola can also be written in "vertex form":
y = a * ( x – h ) ^ 2 + k
Where point (h, k) is the vertex.
You can see how this relates to the standard equation by multiplying it out:
y = a ( x – h ) * ( x – h ) + k
y = a * ( x ^ 2  2 * x * h + h ^ 2 ) + k
y = a x ^ 2 – 2 * a * h * x + a * h ^ 2 + k
The coefficient of x here is –2 a h.
This means that in the standard form ;
y = a * x ^ 2 + b * x + c
the expression
 b / 2 a
gives the x  coordinate of the vertex.
In this case  b / 2 a = 0
that means b = 0 so equation of a parabola is :
y = a x ^ 2 + c
for x = 0 y = 25
25 = a * 0 ^ 2 + c
25 = c
c = 25
for x =  5 y = 0
0 = a * (  5 ) ^ 2 + c
0 = 25 a + 25
 25 a = 25 Divide both sides by  25
a = 25 /  25
a =  1
Also for x = 5 y = 0
0 = a * 5 ^ 2 + c
0 = 25 a + 25
 25 a = 25 Divide both sides by  25
a = 25 /  25
a =  1
Equation of this parabola is :
y =  x ^ 2 + 25 
Answer A.

If you want to see graph go on:
rechneronline.de
In blue rectangle type :
 x ^ 2 + 25
Set :
Range xaxis from  10 to 10
Range xaxis from  10 to 40
and click option Draw