Question

1. (02.08) derive the equation of the parabola with a focus at (0, 1) and a directrix of y = -1. (2 points) f(x)=-\frac{1}{4}x^{2} f(x)=\frac{1}{4}x^{2} f(x)= - 4x^{2} f(x)=4x^{2}

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points $(x,y)$ that are equidistant from the focus $(0,1)$ and the directrix $y = - 1$. The distance between a point $(x,y)$ and the point $(0,1)$ is $\sqrt{(x - 0)^2+(y - 1)^2}$, and the distance between the point $(x,y)$ and the line $y=-1$ is $|y+1|$.

Step2: Set up the distance - equality equation

$\sqrt{(x - 0)^2+(y - 1)^2}=|y + 1|$. Square both sides to get rid of the square - root: $x^{2}+(y - 1)^{2}=(y + 1)^{2}$.

Step3: Expand the equation

Expand the left - hand side: $x^{2}+y^{2}-2y + 1$, and the right - hand side: $y^{2}+2y+1$. So, $x^{2}+y^{2}-2y + 1=y^{2}+2y + 1$.

Step4: Simplify the equation

Subtract $y^{2}+1$ from both sides of the equation: $x^{2}-2y=2y$. Then, move the terms involving $y$ to one side: $x^{2}=4y$, or $y=\frac{1}{4}x^{2}$. So, $f(x)=\frac{1}{4}x^{2}$.

Answer:

B. $f(x)=\frac{1}{4}x^{2}$