Question

identify the focus, directrix, and axis of symmetry of the parabola ( y=\frac{1}{8}x^{2} ).
focus: ( (square,square) )
directrix ( y=square )
axis of symmetry: ( square )-axis
which is the graph of the parabola?

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola is \(x^{2}=4py\). For the given parabola \(y = \frac{1}{8}x^{2}\), we can rewrite it as \(x^{2}=8y\). Comparing with \(x^{2}=4py\), we get \(4p = 8\), so \(p = 2\).

Step2: Find the focus

For a parabola \(x^{2}=4py\), the focus is at \((0,p)\). Since \(p = 2\), the focus is \((0,2)\).

Step3: Find the directrix

For a parabola \(x^{2}=4py\), the directrix is \(y=-p\). Since \(p = 2\), the directrix is \(y = - 2\).

Step4: Find the axis of symmetry

For a parabola \(x^{2}=4py\), the axis of symmetry is the \(y\) - axis (or \(x = 0\)).

Answer:

focus: \((0,2)\), directrix: \(y=-2\), axis of symmetry: \(y\) - axis.