Problem 26 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ x^{2}+y^{2}=25 \quad \text { Up } 3, \text { left } 4 $$

Step-by-Step Solution

Answer

The graph shifts left 4 units and up 3 units to \((x + 4)^2 + (y - 3)^2 = 25\).

Step 1: Identify the transformation

We need to shift the graph of the given equation. The transformation "Up 3, left 4" means we will translate the graph by moving it 4 units to the left and 3 units upwards.

Step 2: Recognize the original equation

The original equation is a circle centered at the origin (0,0) with a radius of 5, because it is in the form \(x^2 + y^2 = r^2\) where \(r = 5\).

Step 3: Apply the horizontal shift

For a horizontal shift to the left by 4 units, replace \(x\) with \((x + 4)\) in the equation. This gives us \((x + 4)^2 + y^2 = 25\).

Step 4: Apply the vertical shift

For a vertical shift upwards by 3 units, replace \(y\) with \((y - 3)\) in the equation. This gives us \((x + 4)^2 + (y - 3)^2 = 25\).

Step 5: Sketch the graphs

The original graph is a circle centered at (0, 0) with a radius of 5. The shifted graph will now be a circle centered at (-4, 3) still with a radius of 5. Plot both circles on the same set of axes and label each with its corresponding equation.

Key Concepts

Circle Equation

A circle is a simple yet meaningful shape in geometry. Its equation helps us understand its fundamental properties such as its center and radius.
The general equation of a circle in a coordinate plane is given by \((x - h)^2 + (y - k)^2 = r^2\).
Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius.
This equation shows us how every point \((x, y)\) that lies on the circle is equidistant from its center. Because of its symmetry, working with circles involves understanding shifts and transformations, which can change its position in the plane without altering its shape or size.

Translation

Translation in geometry is a transformation that slides every point of a shape a specific distance in a given direction.
This transformation doesn't alter the shape or size of the object, just its position.
For circles, translating involves changing the coordinates of its center. This is done by adding or subtracting values to the \(x\) and \(y\) coordinates of the center.

• Moving up or down adjusts the \(y\) value.
• Moving left or right adjusts the \(x\) value.

For example, a translation "up 3, left 4" involves subtracting 4 from the \(x\) coordinate, and adding 3 to the \(y\) coordinate of the circle's center.
Each transformation step requires understanding how these changes affect the placement of the circle but not its inherent geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric shapes in a plane using numerical coordinates.
This branch of mathematics emphasizes the use of algebra to solve geometric problems by defining points using \((x, y)\) pairs.
This capability enables the transformation of shapes like circles accurately using algebraic equations.

• The center of the circle is denoted by its coordinates \((h, k)\).
• It uses precise rules to predict how changes will affect the graph.

By applying transformations correctly, you can shift entire shapes without distorting them, showing the power of coordinate geometry in visualizing and manipulating geometric shapes.

Equation of a Circle

The equation of a circle is paramount for defining its precise location and size in a coordinate plane.
In this exercise, the original circle equation given was \(x^2 + y^2 = 25\), showing the following characteristics:

• The center at the origin \((0,0)\), since it's written in the standard form \((x - 0)^2 + (y - 0)^2 = r^2\).
• Radius of 5, derived from \(r^2 = 25\).

After applying the transformation, "Up 3, left 4", the new equation becomes \((x + 4)^2 + (y - 3)^2 = 25\).
This correctly reflects the circle's new center at \((-4,3)\) with the radius unchanged. Understanding these manipulations provides a clear path from simple algebraic expressions to tangible geometric changes.