Solve the problem.
-Write an equation in standard form of the parabola that has the same shape as the graph of $f ( x ) = - 7 x ^ { 2 }$, but which has a maximum of 9 at $x = 2$.
A) $f ( x ) = - 7 ( x - 2 ) ^ { 2 } + 9$
B) $f ( x ) = - 7 ( x + 2 ) ^ { 2 } + 9$
C) $f ( x ) = 7 ( x - 2 ) ^ { 2 } + 9$
D) $f ( x ) = - 7 ( x - 2 ) ^ { 2 } - 9$

Question

Quizplus · Accepted Answer

The standard form equation of a parabola is given as:
$$f(x) = a(x-h)^2+k$$
where (h,k) represents the vertex of the parabola.
From the given information, the original parabola has the equation $$f(x) = -7x^2$$ which means that the vertex is at (0,0) and it opens downwards.
To shift this parabola such that it has a maximum of 9 at x=2, we need to move the vertex of the parabola to the right by 2 units and up by 9 units. Thus, the new vertex will be at (2,9).
Therefore, the equation of the new parabola is given as:
$$f(x) = a(x-2)^2+9$$
We can compare this with the standard form and see that the value of a will be negative since the parabola opens downwards. We can find the value of a by using the fact that the original parabola has the same shape as the new parabola. Thus, the ratio of the coefficients of the x^2 terms should be the same for both.
$$\frac{a}{-7} = 1 \Rightarrow a=-7$$
Thus, the equation of the new parabola in standard form is:
$$f(x) = -7(x-2)^2+9$$
which matches with choice A.

Solve the Problem f(x)=−7x2f ( x ) = - 7 x ^ { 2 }f(x)=−7x2

Solve the Problem $f ( x ) = - 7 x ^ { 2 }$