Problem 13 - Thomas Calculus

Question

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ y=4 x^{2} $$

Step-by-Step Solution

Answer

Focus: \((0, \frac{1}{16})\), Directrix: \(y = -\frac{1}{16}\).

Step 1: Identifying the Form

The given equation is \( y = 4x^2 \). This is in the form \( y = ax^2 \), which represents a parabola that opens upwards or downwards on the coordinate plane.

Step 2: Finding the Standard Form Parameters

For \( y = ax^2 \), the standard form is \( (x - h)^2 = 4p(y - k) \). Comparing \( y = 4x^2 \) with \( (x - h)^2 = 4p(y - k) \), we rearrange to get \((x-0)^2 = \frac{1}{4}(y-0)\), identifying \( h = 0 \), \( k = 0 \), and \( 4p = \frac{1}{4} \).

Step 3: Calculating the Value of "p"

Solve for \( p \) using the equation \( 4p = \frac{1}{4} \). Thus, \( p = \frac{1}{16} \).

Step 4: Identifying the Focus

For parabolas opening upwards/downwards, the focus is \( (h, k + p) \). So, for \( y = 4x^2 \), the focus is \( (0, 0 + \frac{1}{16}) = (0, \frac{1}{16}) \).

Step 5: Identifying the Directrix

The directrix of the parabola is \( y = k - p \). Therefore, the directrix is \( y = 0 - \frac{1}{16} = -\frac{1}{16} \).

Step 6: Sketching the Parabola

Plot the point \((0, \frac{1}{16})\) as the focus. Draw the horizontal line \(y = -\frac{1}{16}\) as the directrix. Sketch a U-shaped parabola, symmetrical with respect to the y-axis, opening upwards.

Key Concepts

Focus and Directrix

A parabola is defined by its unique feature called the focus, and a line called the directrix. These components help describe the geometrical properties of the parabola.
The focus of a parabola is a point from which distances to any point on the parabola are measured.
The directrix is a line, perpendicular to the axis of symmetry, and serves the same purpose of measurement.

• The focus is one step in the positive direction along the axis of symmetry from the vertex (if the parabola opens upwards or downwards). In this instance of the equation, the focus is at (0, \(\frac{1}{16}\)).
• The directrix, on the other hand, is equally distanced in the opposite direction, resulting in a line for this example at \(y = -\frac{1}{16}\).

Establishing the placement of the focus and directrix is vital for graphing as they guide how the parabola curves, ensuring a consistent distance relationship between the parabola, the focus, and the directrix.

Standard Form of Parabola

Understanding the standard form of a parabola is crucial as it simplifies identifying its key features. The standard form of a parabola is expressed in this way:

• The horizontal form is \((y - k)^2 = 4p(x - h)\),
• The vertical form is \((x - h)^2 = 4p(y - k)\).

When given an equation like \(y = 4x^2\), it reflects a vertically oriented parabola. Here, by rewriting it in standard form to \((x-0)^2 = \frac{1}{4}(y-0)\), we can easily identify:

• \(h = 0\)
• \(k = 0\)
• And importantly, '\(p\)' is \(\frac{1}{16}\), derived from \(4p = \frac{1}{4}\).

By determining these parameters, we set the stage for drafting the parabola's focus and directrix, and subsequently, sketching the parabola.

Graphing Parabolas

Graphing a parabola might seem intimidating, but by breaking it down into steps driven by its equation, it becomes manageable. The key to successfully sketching a parabola is in understanding its symmetry and how it relates to both the focus and the directrix.To graph \(y = 4x^2\):

• Start by plotting the vertex, which is at the origin \((0,0)\) in this equation.
• Locate the focus at \((0, \frac{1}{16})\) and mark it on your graph paper.
• Draw the directrix, a line at \(y = -\frac{1}{16}\), beneath the vertex.
• Use the position of the focus and directrix to sketch the parabola. It will be U-shaped and symmetrical, curving upwards above the directrix.

This approach helps visualize how the parabola 'bends' around its axis of symmetry, which, in this scenario, is the y-axis. Practice with these tools will aid not only in sketching, but also in understanding how parabolas are structured and behave.