Problem 70 - A Graphical Approach to College Algebra

Question

Give the focus, directrix, and axis of each parabola. $$x^{2}=\frac{1}{9} y$$

Step-by-Step Solution

Answer

Focus: \((0, \frac{1}{36})\), Directrix: \(y = -\frac{1}{36}\), Axis: \(x = 0\).

Step 1: Identify the Parabola's Orientation

The given equation is \(x^2 = \frac{1}{9}y\), which is in the form of \(x^2 = 4py\). This indicates that the parabola opens either upwards or downwards.

Step 2: Find the Value of \(p\)

The standard form equation \(x^2 = 4py\) helps us find \(p\) by comparing with \(x^2 = \frac{1}{9}y\). Comparing both equations, we have \(4p = \frac{1}{9}\). Solving for \(p\), we get \(p = \frac{1}{36}\).

Step 3: Determine the Focus

For a parabola in the form \(x^2 = 4py\) with the vertex at the origin, the focus is at \((0, p)\). Thus, with \(p = \frac{1}{36}\), the focus is located at \(\left(0, \frac{1}{36}\right)\).

Step 4: Determine the Directrix

The directrix of a parabola \(x^2 = 4py\) is a line given by \(y = -p\) if the parabola opens upwards. Thus, the directrix is \(y = -\frac{1}{36}\).

Step 5: Identify the Axis of Symmetry

The axis of symmetry for parabolas of the form \(x^2 = 4py\) is the \(y\)-axis, which means the equation of the axis is \(x = 0\).

Key Concepts

Focus of a Parabola

The focus of a parabola holds a vital spot in understanding its structure. Think of the focus as a special point within the curve. All points on a parabola are equidistant from the focus and the directrix. For the given parabola equation, \(x^2 = \frac{1}{9}y\), its focus can be located using the equation, \(x^2 = 4py\). This standard form helps us determine \(p\), the distance from the vertex to the focus.

• First, identify \(4p\) in the equation, which corresponds to \(\frac{1}{9}\).
• When you solve for \(p\), you find \(p = \frac{1}{36}\).
• This means that the focus is at point \((0, \frac{1}{36})\).

The focus lies on the parabola's axis of symmetry, showing its logical link with the other elements.

Directrix of a Parabola

The directrix of a parabola is like a guide line that runs parallel to the parabola's orientation. It's not part of the parabola itself but serves as a geometrical reference used to define the curve.

• For the equation \(x^2 = \frac{1}{9}y\), this line is horizontal because the parabola opens upwards.
• The directrix is determined by the formula \(y = -p\).
• With \(p = \frac{1}{36}\), the directrix becomes \(y = -\frac{1}{36}\).

This equation shows that the directrix is a horizontal line situated below the vertex of the parabola.

Axis of Symmetry

The axis of symmetry is a line of perfect balance for parabolas, defining how they reflect around a central line. For a parabola in the form \(x^2 = 4py\), this axis is vertical.

• It runs through the vertex and every point along this axis reflects onto the opposite side of the parabola.
• In our given equation, the axis of symmetry is represented by \(x = 0\).

Understanding the axis of symmetry aids in graphing and examining the parabola’s symmetrical properties. It means every segment of the curve on one side will have an equal counterpart on the other.