Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$y^{2}=4 x$$

Step-by-Step Solution

Answer

Focus: (1, 0); Directrix: x = -1

Step 1: Identify the Form of the Parabola

The given equation of the parabola is \(y^2 = 4x\). This is in the standard form \(y^2 = 4px\), where the parabola opens to the right and the vertex is at the origin \((0, 0)\).

Step 2: Determine the Value of p

Compare the given equation \(y^2 = 4x\) with the standard form \(y^2 = 4px\) to find \(p\). Here, \(4p = 4\), so \(p = 1\).

Step 3: Find the Focus

For a parabola \(y^2 = 4px\) that opens to the right, the focus is at \((p, 0)\). Given \(p = 1\), the coordinates of the focus are \((1, 0)\).

Step 4: Determine the Equation of the Directrix

The directrix of a parabola \(y^2 = 4px\) that opens to the right is given by \(x = -p\). With \(p = 1\), the equation of the directrix is \(x = -1\).

Step 5: Sketch the Parabola, Focus, and Directrix

Draw the parabola with its vertex at the origin (0, 0) and opening to the right. Plot the focus at point \((1, 0)\) and draw the vertical line of the directrix at \(x = -1\). Ensure the parabola curves wide enough to emphasize its shape relative to the focus and directrix.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola is essential to know when solving for its characteristics like the focus and directrix. It provides a formula that is used to derive these components. For parabolas that open horizontally or vertically, the standard form is given by two primary equations:

• Horizontal opening: \( y^2 = 4px \)
• Vertical opening: \( x^2 = 4py \)

Here, \( p \) represents the distance between the vertex of the parabola and the focus. In our specific exercise, the equation \( y^2 = 4x \) is perfectly aligned with the standard form of a horizontally opening parabola, \( y^2 = 4px \). This means the parabola opens to the right with its vertex located at the origin \((0, 0)\). Comparing \( y^2 = 4x \) with \( y^2 = 4px \) shows that \( p \) is equal to 1, providing the foundation for finding the focus and directrix.

Focus of a Parabola

The focus of a parabola is a key point that significantly influences its geometric properties. It is a single point interior to the parabola that helps define its shape. For a parabola in the standard form \( y^2 = 4px \), which opens horizontally, the focus is located at \((p, 0)\).
To find the focus of a parabola, identify the value of \( p \) from its equation. In the given equation \( y^2 = 4x \), we have determined \( p = 1 \). Thus, the focus of this parabola is precisely at the coordinates \((1, 0)\). The parabola curves around this point, and any point on the parabola is equidistant to the focus and the directrix.

Directrix of a Parabola

The directrix of a parabola is a line that serves as a reference for defining the parabola's shape along with its focus. For a horizontally opening parabola with the equation \( y^2 = 4px \), the directrix is a vertical line given by \( x = -p \).
Comparing \( y^2 = 4x \) with the standard form reveals \( p = 1 \), making the equation of the directrix \( x = -1 \). This means the directrix is a vertical line located one unit to the left of the vertex.
Every point on the parabola is equidistant from the focus and this directrix, which is a critical property that defines the balanced shape of the parabola. Visualizing this helps in sketching and understanding how the parabola interacts with both its focus and its directrix.