Question

Identify the vertex, focus, and directrix of the graph. Which of the following equations represents the parabola in the graph?

Answer 1

Answer:

Equation: $(x-2)^2=16(y-2)$

Vertex: $(2,2)$

Directrix: $y=-2$

Focus: $(2,6)$

Step-by-step explanation:

Standard Form of Vertical Parabola:

• Equation -> $(x-h)^2=4p(y-k)$ where $p\neq 0$
• Vertex -> $(h,k)$
• Directrix -> $y=k-p$
• Focus -> $(h,k+p)$

We know that our vertex is $(h,k)$ -> $(2,2)$, therefore, we can determine the value of $p$ by selecting a point from the parabola as $(x,y)$:

$(x-h)^2=4p(y-k)$

$(x-2)^2=4p(y-2)$

$(6-2)^2=4p(3-2)$

$(4)^2=4p(1)$

$16=4p$

$4=p$

Therefore:

• Directrix -> $y=k-p$ -> $y=2-4$ -> $y=-2$

• Focus -> $(h,k+p)$ -> $(2,2+4)$ -> $(2,6)$

Conclusion:

• Equation -> $(x-2)^2=16(y-2)$
• Vertex -> $(2,2)$
• Directrix -> $y=-2$
• Focus -> $(2,6)$

Review the graph for a visual