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
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 x-axis from - 10 to 10
Range x-axis from - 10 to 40
and click option Draw
To determine the equation of the parabola, we need to use the vertex form of a parabola equation, which is given by:
f(x) = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
In this case, the given vertex is (0, 25). Therefore, the equation becomes:
f(x) = a(x - 0)^2 + 25
f(x) = a(x^2) + 25
Next, we need to find the value of 'a'. To do that, we can use the given zeros (x = -5 and x = 5). Since the zeros are the x-intercepts where f(x) equals zero, we can substitute these values into the equation and solve for 'a'.
When x = -5:
0 = a((-5)^2) + 25
0 = 25a + 25
25a = -25
a = -1
Now that we have the value of 'a', we can substitute it back into the equation:
f(x) = -1(x^2) + 25
f(x) = -x^2 + 25
Comparing this equation to the given options, we can see that the correct answer is (D) x^2 - 25.