Problem 46 - Thomas Calculus

Question

Exercises \(45-48\) give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. $$ y^{2}=-12 x, \quad \text { right } 4, \text { up } 3 $$

Step-by-Step Solution

Answer

The new equation is \( (y - 3)^2 = -12(x - 4) \) with vertex (4, 3), focus (1, 3), and directrix \( x = 7 \).

Step 1: Identify the Standard Form

The given equation is \( y^2 = -12x \). This is a horizontal parabola in the form \( y^2 = 4px \), where \( p < 0 \) (as it's opening left). The value of \( 4p = -12 \), so \( p = -3 \). Hence, the vertex of this parabola is at \((0, 0)\) and its focus is \((-3, 0)\). The equation of the directrix is \( x = 3 \).

Step 2: Determine New Vertex

The problem states that the parabola is to be shifted 4 units to the right and 3 units up. Therefore, the new vertex for the parabola is \((4, 3)\).

Step 3: Apply Shifts to Equation

Since the parabola is shifted 4 units right and 3 units up, substitute \( x \) with \( x' - 4 \) and \( y \) with \( y' - 3 \) in the original equation. This gives us: \[ (y' - 3)^2 = -12(x' - 4) \].

Step 4: Simplify the New Equation

Expanding the equation from the previous step results in:\[ y'^2 - 6y' + 9 = -12x' + 48 \]. Rearranging terms to match the standard form gives us: \[ y'^2 - 6y' = -12x' + 39 \].

Step 5: Calculate New Focus

The new focus is found by applying the same shifts to the original focus point. Thus, the new focus is \((-3 + 4, 0 + 3) = (1, 3)\).

Step 6: Determine New Directrix

The directrix is shifted by the same amount as the focus. The original directrix \( x = 3 \) becomes \( x = 3 + 4 = 7 \).

Key Concepts

Parabolas

A parabola is a specific type of conic section, which is a curve formed at the intersection of a cone with a plane. Parabolas have a distinct U-shaped curve. One of the defining features of a parabola is that any point on it is equidistant from a specific point, called the focus, and a line, called the directrix. This property of parabolas can be useful for computations and constructions in geometry and algebra.
Parabolas can open upwards, downwards, left, or right. The direction they open depends on the form of their equation. For instance, a parabola represented by the equation \( y^2 = 4px \) opens horizontally, whereas \( x^2 = 4py \) indicates a vertical orientation. This exercise deals with a horizontally opening parabola.

Vertex

The vertex of a parabola is the point where it turns or changes direction. It's the midpoint between the directrix and the focus. For parabolas given in the simplest form, the vertex often starts at the origin \((0, 0)\).
In our exercise, the original parabola has its vertex at \((0, 0)\). However, after applying the given transformations—4 units to the right and 3 units up—the vertex shifts to the new position \((4, 3)\).
This coordinate change corresponds to rewriting the parabola’s equation to match the new shifted position, ensuring the fundamental properties relative to the vertex and axis of symmetry are maintained.

Focus

The focus of a parabola is a fixed point used to define the curve. Each point on the parabola is equidistant from the focus and the directrix. The position of the focus plays a crucial role in determining the parabola's steepness and direction.
In our example, before shifting, the focus is located at \((-3, 0)\). Applying the transformation (right 4, up 3) translates it to the coordinates \((1, 3)\).
This new focus means that every point on the transformed parabola maintains its equal distance property, now measured from \((1, 3)\) rather than the original \((-3, 0)\). Understanding the focus's role helps understand the geometric and algebraic shape of the parabola.

Directrix

The directrix of a parabola is a line perpendicular to the axis of symmetry and helps to define the curve given its relationship with the focus. The directrix and the focus are equidistant from any point on the parabola.
In the original parabola, the directrix is the vertical line \(x = 3\). After shifting the parabolic graph according to the instructions, the line also increases by the same horizontal distance, resulting in the new directrix \(x = 7\).
The consistent relationship with the focus ensures that not only does the parabola maintain its geometric shape, but it also preserves uniformity in its properties across any transformations applied.