Problem 65 - Algebra 2

Question

A parabola with vertex at the origin and focus at \((-1,0)\) is translated 3 units to the right and 4 units up. What is the equation of the translated parabola? Show your work.

Step-by-Step Solution

Answer

The equation of the translated parabola is \(x^2 - 6x - 4y + 7 = 0\).

Step 1: Find Original Parabola

A parabola is defined by its vertex and focus. The vertex is at the origin and the focus is at (-1,0). The standard equation of a parabola is \(x^2 = 4py\), where p is the distance from the vertex to the focus. Here, p = -1, hence the equation of the parabola is \(x^2 = -4y\).

Step 2: Apply Translations

A horizontal translation by h units and a vertical translation by k units has the effect of replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) in the equation of the original function. Since the parabola is being translated 3 units right and 4 units up, the new x becomes \(x-3\) and the new y becomes \(y-4\). Substituting these into the original equation, we have \((x-3)^2 = -4(y - 4)\).

Step 3: Simplify Equation

Simplify the equation by expanding \((x-3)^2\) and distributing -4. This results in the equation \(x^2 - 6x + 9 = -4y + 16\). Rearranging terms, we then have \(x^2 - 6x - 4y + 7 = 0\). This is the equation of the translated parabola.

Key Concepts

Vertex and Focus

In parabola equations, the vertex and focus play a critical role in determining the shape and orientation of the curve. The vertex is the point where the parabola changes direction, and the focus is a point used to define how the parabola spreads. For a parabola where the vertex is at the origin

• The focus is another crucial point located either along the x-axis or y-axis depending on the parabola's orientation.
• The focus is used to find the value of "p", which defines the "tightness" of the parabola.
• In the provided exercise, the vertex is (0,0), and the focus is at (-1,0), indicating a horizontal parabola.

This means the parabola opens sideways rather than upwards or downwards as in the common equations, such as \(y^2 = 4px\). The distance "p" is calculated from the vertex to the focus. Since the focus is one unit to the left of the vertex in this scenario, \(p = -1\). Thus, the standard form of this parabola's equation is \(x^2 = -4y\).

Coordinate Translation

When translating a parabola in the coordinate plane, we are essentially "shifting" its position. The process involves moving the entire graph without changing its shape. Translation refers to the horizontal and vertical shifts applied to the original equation.

• A horizontal shift of \(h\) units is applied by replacing \(x\) with \(x-h\).
• A vertical shift of \(k\) units is applied by replacing \(y\) with \(y-k\).

For example, in our exercise, the parabola is shifted 3 units to the right and 4 units up. This means:

• Every "x" in the equation of the parabola is replaced with \(x-3\).
• Every "y" is replaced with \(y-4\).

These replacements alter the position of the parabola while maintaining its curvature and opening direction. Applying these changes to the original equation \(x^2 = -4y\), the translated equation becomes \((x-3)^2 = -4(y-4)\).

Parabola Equation Transformation

To obtain the final equation of the translated parabola, further manipulate the terms in the equation to simplify or expand them. This involves algebraic manipulation to arrive at a standard form.

• Begin by expanding \((x-3)^2\) to get \(x^2 - 6x + 9\).
• Distribute -4 across \(y - 4\) to get \(-4y + 16\).
• Combine these into a single equation to reveal: \(x^2 - 6x + 9 = -4y + 16\).

Next, rearrange the terms to represent the parabolic equation in a neat form. Move all terms to one side of the equation to get:

• \(x^2 - 6x - 4y + 7 = 0\). This step harmonizes the transformed parabola equation for further analysis or graphing.

Equation transformation is crucial as it clearly exhibits the parabola's new position following translation, making analysis straightforward.