Find the quadratic function $f$ whose graph intersects the $x$-axis at $( 2,0 )$ and $( 3,0 )$ and the $y$-axis at $( 0 , - 18 )$.
A) $f ( x ) = 3 x ^ { 2 } + 3 x + 9$
B) $f ( x ) = - 3 x ^ { 2 } + 15 x - 18$
C) $f ( x ) = - 3 x ^ { 2 } - 3 x + 6$
D) $f ( x ) = 3 x ^ { 2 } - 3 x - 18$
E) $f ( x ) = 3 x ^ { 2 } - 15 x - 18$

Question

Quizplus · Accepted Answer

The quadratic function $f(x)$ can be found using the fact that it intersects the $x$-axis at $(2,0)$ and $(3,0)$, which means $x=2$ and $x=3$ are its roots. Therefore, $f(x)$ can be expressed as $f(x) = a(x-2)(x-3)$. Since it intersects the $y$-axis at $(0,-18)$, we have $f(0) = -18$, which allows us to solve for $a$. Substituting $x=0$ into $f(x) = a(x-2)(x-3)$ gives $a(0-2)(0-3) = -18$, leading to $6a = -18$, so $a = -3$. Thus, $f(x) = -3(x-2)(x-3) = -3(x^2 - 5x + 6) = -3x^2 + 15x - 18$.

Find the Quadratic Function fff Whose Graph Intersects The xxx -Axis At (2,0)( 2,0 )(2,0)

Find the Quadratic Function $f$ Whose Graph Intersects The $x$ -Axis At $( 2,0 )$