Question

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. 25-x^2

B. 1- 1/25 x^2

C. x^2-1

D. x^2-25

Answer 1

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 2

Answer A.

Answer 3

If you want to see graph go on:

rechneronline.de

In blue rectangle type :

- x ^ 2 + 25

Set :

Range x-axis from - 10 to 10

Range x-axis from - 10 to 40

and click option Draw