Problem 25 - 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. $$ x^{2}+y^{2}=49 \quad \text { Down } 3, \text { left } 2 $$

Step-by-Step Solution

Answer

Shift the center from (0,0) to (-2,-3), resulting in \((x+2)^2+(y+3)^2=49\).

Step 1: Identify the Original Equation

The original equation is given as \( x^2 + y^2 = 49 \). This represents a circle centered at (0,0) with a radius of 7.

Step 2: Determine the Translations

We need to shift the graph of the circle down by 3 units and left by 2 units. This means we adjust the center of the circle from (0,0) to (-2,-3).

Step 3: Write the Equation for the Shifted Graph

The formula for a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center. After shifting, the new equation is: \[(x+2)^2 + (y+3)^2 = 49\]

Step 4: Sketch the Graphs

To sketch, first draw the original circle centered at (0,0) with radius 7. Then, draw the shifted circle centered at (-2,-3) with the same radius of 7. Label the original graph as \(x^2+y^2=49\) and the shifted graph as \((x+2)^2+(y+3)^2=49\).

Step 5: Labeling and Comparison

Ensure both graphs are clearly labeled. The original graph, \(x^2 + y^2 = 49\), should show as a circle centered at (0,0). The shifted graph, \((x+2)^2 + (y+3)^2 = 49\), should appear as a circle shifted 3 units down and 2 units left.

Key Concepts

Graph Translations

Understanding how to translate the graph of a circle is essential when moving its position on a coordinate plane. A translation is simply a move without any rotation or resizing of the graph. For a circle, this means shifting the entire graph horizontally, vertically, or both. The key is to adjust the coordinates of the circle's center while maintaining its shape and size.

In the original problem, we started with the equation \(x^2 + y^2 = 49\). This represents a circle with its center at the origin (0,0) and a radius of 7. A translation can adjust the center to a new point. In this case, the circle is moved 3 units down and 2 units to the left. This results in the circle's new center being located at (-2,-3).

• Translate left or right by adding or subtracting from the \(x\) coordinate.
• Translate up or down by adding or subtracting from the \(y\) coordinate.

By understanding graph translations, you can easily reposition graphs while keeping their inherent properties intact.

Coordinate Shifts

Coordinate shifts involve changing the position of a figure, like a circle, on a graph. This is achieved by altering the coordinates of its center. For our initial circle, the center at (0,0) needed to be shifted. Here's exactly how:

To shift left or right, we modify the \(x\) coordinate:

• Left: Add a value, making \(x + 2\) in our case.
• Right: Subtract a value, which would look different from our example.

To manage the vertical movement, or shifting up or down, you change the \(y\) coordinate:

• Down: Add to \(y\), as seen in \(y + 3\).
• Up: Subtract from \(y\), which wasn't required here.

These perpendicular moves from horizontal to vertical ensure the center relocates accurately. As such, the circle remains the same size but appears at a different place on the grid.

Equation Transformation

Altering the equation of a circle is a direct result of moving its center, a crucial skill for handling more complex transformations. When a circle's graph is moved, its formula adapts via coordinate changes:

In the formula \((x-h)^2 + (y-k)^2 = r^2\), \((h,k)\) identifies the center. Shifts in position cause \(h\) and \(k\) to differ from their original settings according to the transformations:

• Horizontal Shift: \(h\) changes by the shift amount, such as \(x + 2\) for 2 left.
• Vertical Shift: \(k\) also changes by the shift amount, demonstrated by \(y + 3\) for 3 down.

Utilize the translation information directly to adjust \(h\) and \(k\). Hence, original circles like \(x^2 + y^2 = 49\) evolve into \((x+2)^2 + (y+3)^2 = 49\) post-translation. Transformations don't alter radius \((r)\); instead, they modify position parameters within the equation.