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: (0,1); Directrix: \( y = -1 \).

Step 1: Identify the Parabola Form

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

Step 2: Determine Orientation and Coefficient

Since the equation is in the form \( y = 4x^2 \) and \( a = 4 > 0 \), the parabola opens upwards.

Step 3: Find Vertex Form and Vertex

The equation \( y = 4x^2 \) is already in the vertex form of \( y = a(x-h)^2 + k \). Here, \( h = 0 \) and \( k = 0 \), making the vertex at the origin (0,0).

Step 4: Calculate "p" Value

In a vertical parabola in the form \( y = a(x-h)^2 + k \), the parameter \( p \) is calculated by \( 4p = a \). Thus, \( 4p = 4 \) so \( p = 1 \).

Step 5: Locate the Focus

Since \( p = 1 \) and the vertex is at (0,0), the focus \((0, p)\) is located at (0,1).

Step 6: Determine the Directrix

The directrix is a horizontal line \( y = k - p \). With \( k = 0 \) and \( p = 1 \), the directrix is \( y = -1 \).

Step 7: Sketch the Parabola

Plot the vertex at (0,0), the focus at (0,1), and the directrix as the line \( y = -1 \). Draw the parabola opening upwards from the vertex, equidistant from the focus and directrix.

Key Concepts

Focus

The focus of a parabola is a key point that helps in defining the curve's shape. In general, the focus of a parabola is located inside the curve. It reflects how the parabola opens up and captures any incoming parallel rays to its axis. For the given parabola equation, which is in the form

• \( y = ax^2 \)
• When the vertex is at the origin (0,0), the focus is found using the relation \((0, p)\), where \(4p = a\).

In this exercise, since \( a = 4 \), we calculate that \( p = 1 \). Therefore, the focus is located at (0,1). The focus is above the vertex because the parabola opens upwards. This means any vertical line towards the focus will reflect off the parabolic surface.

Directrix

The directrix of a parabola is a fixed line used to define the curvature's alignment. It serves as a boundary that pairs with the focus to maintain the parabola's specific structure. In the case of a vertical parabola like ours, this line is horizontal. For the equation \( y = 4x^2 \), the directrix can be found by taking the vertex's vertical position \( k \), and subtracting the value \( p \):

• The directrix is given by the equation \( y = k - p \).
• In this case, \( k = 0 \) and \( p = 1 \), resulting in the directrix line \( y = -1 \).

This directrix lies below the vertex when the parabola is upwards opening, indicating a uniform distance between it and the focus.

Vertex Form

The vertex form of a parabola provides a straightforward way to identify its peak or minimum point, also known as the vertex. The standard vertex form is

• \( y = a(x-h)^2 + k \),
• where \((h, k)\) is the vertex.

Since the provided equation \( y = 4x^2 \) already fits this form with \( h = 0 \) and \( k = 0 \), the vertex is at the origin (0,0). In this form, the vertex stands as a reference for determining the other aspects of the parabola, including its symmetry and orientation. It helps in understanding the overall shape and behavior of the curve.

Upwards Opening Parabola

Understanding the direction in which a parabola opens is crucial for graphing and interpreting various scenarios. A parabola is defined as upwards opening if its "arms" extend towards the positive

• y-axis,
• meaning \( a > 0 \) in the equation \( y = ax^2 \).

In our specific case, \( y = 4x^2 \), the coefficient \( a = 4 \) is greater than zero, signifying an upwards opening parabola. This direction is critical, as it affects both the focus and directrix positions: the focus is above the vertex, and the directrix is below it. Visualization of these relationships helps in sketching the parabola accurately.