Problem 38 - Algebra 2

Question

The graph of each equation is to be translated 3 units right and 5 units up. Write each new equation. \(\frac{(x-3)^{2}}{64}+\frac{(y+3)^{2}}{36}=1\)

Step-by-Step Solution

Answer

The new equation for the ellipse after translation 3 units to the right and 5 units up is \(\frac{(x-6)^2}{64}+\frac{(y-2)^2}{36}=1\).

Step 1: Understand the Original Ellipse Equation

The given equation represents an ellipse with center (3,-3). The equation of an ellipse in standard form is \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\), where \((h,k)\) are the coordinates of the center. Here, \(h=3\) and \(k=-3\).

Step 2: Translate X and Y coordinates

Translation of the ellipse 3 units to the right and 5 units up means moving the center of the ellipse to the point \((3+3, -3+5)\), which is \((6,2)\). So the \(h\) value will be 6 and the \(k\) value will be 2.

Step 3: Substitute new center into the equation

Substitute \(h=6\) and \(k=2\) into the standard form of the ellipse equation. The new equation after translation becomes \(\frac{(x-6)^2}{64}+\frac{(y-2)^2}{36}=1\). This is the new equation for the ellipse after it has been translated 3 units to the right and 5 units up.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects arithmetic and geometry using coordinates. It allows us to express geometric shapes, like lines, circles, and ellipses, using equations on a coordinate plane.
In the context of the ellipse, coordinate geometry helps locate points and understand their relationships with the shape. The coordinates (h, k) are pivotal as they indicate the center of the ellipse, and changes to these coordinates result in translation of the ellipse itself.
Utilizing coordinate geometry, transformations like translations are straightforward by simply altering the coordinates. This helps one construct new geometric figures with ease without detailed geometric construction.

Ellipse Equation

An ellipse is a geometric shape characterized by its symmetry and smooth, oval-like profile. Its equation in coordinate geometry is represented in the standard form as \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \].
Here,

• (h, k) denotes the center of the ellipse.
• a is the semi-major axis's length, indicating the ellipse's extension in the horizontal direction if longer.
• b is the semi-minor axis's length, denoting the vertical extension.

A typical step in problems is to determine these values correctly from the equation to understand the ellipse's size and orientation.
The original ellipse has a center at (3, -3) with an extensive reach along the x-axis (due to a being larger than b), making it a crucial parameter in determining the ellipse's nature.

Graph Transformation

Graph transformations allow us to alter equations to reflect shifts and changes in shape.
When an ellipse is translated on a graph, its shape and size remain, but its position changes.
In the given example, we translate the ellipse by 3 units right and 5 units up. This means changing the coordinates of the center from (3, -3) to (6, 2).

• Moving 3 units right adds 3 to the x-coordinate.
• Shifting 5 units up adds 5 to the y-coordinate.

Incorporating these changes into the equation leads to a new center, forming the updated ellipse equation.
Ultimately, this transformation results in a new equation: \[ \frac{(x-6)^2}{64} + \frac{(y-2)^2}{36} = 1 \], indicating its new position on the graph.