Write an equation for the quadratic function whose graph contains the given vertex and point.
-Vertex $( 2,4 )$, point $( 5,22 )$
A) $P ( x ) = - 2 x ^ { 2 } - 8 x + 4$
B) $P ( x ) = 2 x ^ { 2 } - 8 x + 12$
C) $P ( x ) = 5 x ^ { 2 } - 8 x - 12$
D) $P ( x ) = 2 x ^ { 2 } - 2 x + 4$

Question

Quizplus · Accepted Answer

The general form of a quadratic function is $P(x) = ax^2 + bx + c$. Given the vertex $(h, k) = (2, 4)$, we can use the vertex form of a quadratic function, which is $P(x) = a(x - h)^2 + k$. Substituting the vertex into this equation gives $P(x) = a(x - 2)^2 + 4$. To find the value of $a$, we use the given point $(5, 22)$. Substituting $x = 5$ and $P(x) = 22$ into the equation gives $22 = a(5 - 2)^2 + 4$, which simplifies to $22 = 9a + 4$. Solving for $a$ gives $a = 2$. Substituting $a = 2$ back into the vertex form equation and expanding it gives $P(x) = 2(x - 2)^2 + 4$, which simplifies to $P(x) = 2x^2 - 8x + 12$, matching option B.

Write an Equation for the Quadratic Function Whose Graph Contains (2,4)( 2,4 )(2,4)

Write an Equation for the Quadratic Function Whose Graph Contains $( 2,4 )$