Question

14.8.2 test (cst): conic sections
the vertex of the parabola below is at the point (3, 2), and the point (4, 6) is on the parabola. what is the equation of the parabola?
a. y = 2(x - 2)^2+3
b. y = 4(x - 3)^2+2
c. y = 4(x + 3)^2-2
d. x = 6(y - 3)^2+2

Explanation:

Step1: Recall parabola vertex - form

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. Given the vertex $(h,k)=(3,2)$, the equation of the parabola is $y=a(x - 3)^2+2$.

Step2: Find the value of $a$

Substitute the point $(x = 4,y = 6)$ into the equation $y=a(x - 3)^2+2$. We get $6=a(4 - 3)^2+2$.
Simplify the right - hand side: $6=a\times1^2+2$, which is $6=a + 2$.
Subtract 2 from both sides: $a=6 - 2=4$.

Step3: Write the final equation

Substitute $a = 4$ into the vertex - form equation. The equation of the parabola is $y=4(x - 3)^2+2$.

Answer:

B. $y = 4(x - 3)^2+2$