Problem 47 - 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. $$ x^{2}=8 y, \quad \text { right } 1, \text { down } 7 $$

Step-by-Step Solution

Answer

New equation: \((x-1)^2 = 8(y+7)\). Vertex: (1,-7). Focus: (1,-5). Directrix: \(y=-9\).

Step 1: Understand the original equation

The given equation of the parabola is \(x^2 = 8y\). This is a standard form of a parabola opening upwards, with its vertex at the origin (0,0).

Step 2: Analyze the parabola's shift

The problem specifies a shift to the right by 1 unit and down by 7 units. This means the vertex of the parabola will move from (0,0) to (1,-7).

Step 3: Apply the horizontal and vertical shifts

For a parabola with an equation \(x^2 = 4py\), shifting right by 1 unit and down by 7 units modifies it to \((x-1)^2 = 4p(y+7)\). Our \(8y\) term implies \(4p = 8\), so \(p = 2\). Thus, the new equation becomes \((x-1)^2 = 8(y+7)\).

Step 4: Determine the new vertex

After applying the shifts, the new vertex is located at (1,-7). This is based on the fact that we moved right 1 unit and down 7 units from the origin.

Step 5: Find the new focus

For the original equation \(x^2 = 8y\), the focus is at (0,2) (since \(p=2\)). Shifting the focus right 1 unit and down 7 units yields the new focus: (1,-5).

Step 6: Find the new directrix

The original directrix is at \(y = -2\) (since it is \(y = -p\)). When shifted right by 1 unit and down by 7 units, the directrix remains vertical, changing only vertically to \(y = -9\).

Key Concepts

Vertex

Understanding the vertex of a parabola is crucial when dealing with parabola transformations. In general, the vertex of a parabola is the point where the curve changes direction, acting as a turning point. For standard parabolas of the form \(x^2 = 4py\), the vertex is initially at the origin, which is \((0,0)\).

When we apply transformations, such as shifts to the right, left, up, or down, the position of the vertex changes accordingly. For instance, in the exercise provided, the original parabola \(x^2 = 8y\) has its vertex at \((0,0)\).

• Shifting **right by 1 unit** means adding 1 to the x-coordinate.
• Shifting **down by 7 units** subtracts 7 from the y-coordinate.

This results in a new vertex located at \((1, -7)\). Such transformations are typical when finding new positions of a parabola's vertex.

Focus

The focus of a parabola is a fixed point used in its geometric definition, lying inside the curve, where the parabola "focuses" all its collected paths.In the case of the equation \(x^2 = 4py\), the focus of the corresponding parabola is originally at \((0,p)\), where \(p\) is the distance from the vertex to the focus inside the parabola. From the given exercise, with \(x^2 = 8y\), \(p = 2\), so the original focus is at \((0,2)\).

Like the vertex, the focus changes position when the parabola is transformed.

• Shifting **right by 1 unit** means adding 1 to the x-component of the focus.
• Shifting **down by 7 units** involves subtracting 7 from the y-component.

After applying these shifts, the new focus is \((1, -5)\). Understanding how to find the focus after transformations helps visualize and graph the parabola accurately.

Directrix

Another fundamental part of a parabola is its directrix, a line situated opposite the focus. The parabola maintains its form in such a way that the perpendicular distance from any point on the parabola to the focus equals the perpendicular distance from that point to the directrix. For the original equation \(x^2 = 8y\), the directrix is initially at \(y = -2\), given by the equation \(y = -p\).

When we apply shifts to the parabola, the directrix generally shifts parallel to itself.

• While a shift right doesn't change its equation form because directrix is horizontal,
• shifting **down by 7 units** just subtracts 7 from the directrix value.

This results in a new directrix at \(y = -9\). Recognizing how directrix shifts play a crucial role in correctly transforming and understanding the parabola's behavior.