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 can be found using the fact that it intersects the $x$-axis at $(2,0)$ and $(3,0)$, meaning $x = 2$ and $x = 3$ are its roots. Thus, the function can be expressed as $f(x) = a(x - 2)(x - 3)$. Since it intersects the $y$-axis at $(0, -18)$, when $x = 0$, $f(x) = -18$. Substituting $x = 0$ into the equation gives $a(-2)(-3) = -18$, leading to $a = -3$. Therefore, the function is $f(x) = -3(x - 2)(x - 3)$, which simplifies to $f(x) = -3x^2 + 15x - 18$.

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 $( 2,0 )$