Respuesta :
Answer:
Option 4.
Step-by-step explanation:
The vertex form of a parabola is
[tex]g(x)=a(x-h)^2+k[/tex] ... (1)
where, a is a constant (h,k) is vertex.
The given function is
[tex]f(x)=x^2[/tex]
The vertex of the function is (0,0) and it goes through (-2, 4) and (2, 4).
It is given that the vertex of function g(x) is at (5,2).
Substitute h=5 and k=2 in equation.
[tex]g(x)=a(x-5)^2+2[/tex]
g(x) is passes through (3, 6).
[tex]6=a(3-5)^2+2[/tex] .... (2)
[tex]6-2=4a[/tex]
[tex]4=4a[/tex]
Divide both sides by 4.
[tex]a=1[/tex]
Substitute a=1 in equation (2).
[tex]g(x)=1(x-5)^2+2[/tex]
[tex]g(x)=(x-5)^2+2[/tex]
The function g(x) is [tex]g(x)=(x-5)^2+2[/tex].
Therefor, the correct option is 4.
The equation of the considered parabola represented by g is given as:
[tex]g(x) = (x - 5)^2 + 2[/tex]
Thus, Option 4: [tex]g(x) = (x-5)^2 + 2[/tex] is correct.
Given that:
- [tex]f(x) = x^2[/tex]
- g(x) passes through (3,6), (7,6)
- The parabola represented by g(x) has vertex at (5,2)
Explanation and formation of equation:
The parabola [tex]y = a (x-h)^2 + k[/tex] has vertex at [tex](h,k)[/tex]
For our case, putting value of h and k, we get:
[tex]g(x) = a(x - 5)^2 + 2[/tex]
Since the parabola passes through (3,6), thus it must satisfy the equation of parabola. Thus,
[tex]6 = a(3-5)^2 + 2\\6 = a \times 4 + 2\\4 = 4a\\a = 1[/tex]
Thus, equation of the considered parabola represented by g is given as:
[tex]g(x) = (x - 5)^2 + 2[/tex]
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