Find the quadratic function $f$ whose graph intersects the $x$-axis at $( 2,0 )$ and $( 3,0 )$ and the $\mathrm { y }$-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 given points. Since it intersects the $x$-axis at $(2,0)$ and $(3,0)$, it can be written in the form $f(x) = a(x - 2)(x - 3)$. To find $a$, use the $y$-axis intersection point $(0, -18)$, which gives $f(0) = a(0 - 2)(0 - 3) = -18$. Solving for $a$ gives $a = 3$, so the function is $f(x) = 3(x - 2)(x - 3) = 3(x^2 - 5x + 6) = 3x^2 - 15x + 18$. However, the correct sign for the constant term to match the $y$-axis intersection is $-18$, leading to $f(x) = -3x^2 + 15x - 18$, which matches option B.

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 $( 2,0 )$