Problem 19 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be 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

The shifted graph's equation is \((x+2)^2 + (y+3)^2 = 49\).

Step 1: Understand the Equation

The given equation \( x^2 + y^2 = 49 \) represents a circle centered at the origin (0,0) with a radius of 7, because the equation is in the standard form of a circle equation \( (x-h)^2 + (y-k)^2 = r^2 \), where \((h,k)\) is the center and \(r\) is the radius.

Step 2: Determine the Transformation

The transformation involves shifting the circle down by 3 units and left by 2 units. This means we will adjust the center of the circle from (0,0) to (-2,-3).

Step 3: Apply the Shift to the Equation

To shift the circle left by 2 units, we replace \(x\) with \((x+2)\) in the equation. To shift the circle down by 3 units, we replace \(y\) with \((y+3)\) in the equation. Thus, the new equation becomes \((x+2)^2 + (y+3)^2 = 49\).

Step 4: Sketch the Graphs

Sketch the original graph of the circle \(x^2 + y^2 = 49\), which is centered at the origin (0,0). Then sketch the transformed circle \((x+2)^2 + (y+3)^2 = 49\), which is centered at (-2,-3). Be sure to label both circles with their corresponding equations.

Step 5: Label the Shift

Clearly label the original circle with its equation \(x^2 + y^2 = 49\) and the shifted circle with its new equation \((x+2)^2 + (y+3)^2 = 49\) on the graph.

Key Concepts

Equation of a Circle

The equation of a circle is an essential concept in geometry. It's a way to mathematically describe the set of all points that are equidistant from a central point, known as the center of the circle. In its standard form, the equation is written as \((x-h)^2 + (y-k)^2 = r^2\). Here's what each part means:

• \((h,k)\) - This pair represents the coordinates of the circle's center. In other words, it's the exact point from which every other point on the circle is the same distance, called the radius.
• \(r\) - The radius is the distance from the center to any point on the circle. It determines the circle's size. The square of the radius, \(r^2\), is what appears in the equation.

If you look at the given example, \(x^2 + y^2 = 49\), it's easy to see that its center is at \((0,0)\) with a radius of 7, since \(\sqrt{49} = 7\). Understanding this form helps us intuitively grasp the circle's positioning and dimensions in a coordinate plane.

Coordinate Shifts

Coordinate shifts allow us to move geometric figures across the coordinate plane without changing their shape or orientation. A shift involves moving the entire figure up, down, left, or right.

For circles specifically, coordinate shifts are performed by altering the \(h\) and \(k\) values in the circle's equation. A shift can be described as:

• Left or Right: Changing \(h\) shifts the circle left when subtracting and right when adding. For example, replacing \(x\) with \((x+2)\) moves the circle left by 2 units.
• Up or Down: Changing \(k\) shifts the circle up when subtracting and down when adding. For example, replacing \(y\) with \((y+3)\) moves the circle down by 3 units.

In our exercise, the transformation involved moving the circle left by 2, and down by 3, resulting in a new center at \((-2, -3)\) and a new equation: \((x+2)^2 + (y+3)^2 = 49\). This process shows how adjustments in \(h\) and \(k\) create shifts in the circle's position.

Graphing Transformations

Graphing transformations involve altering a geometric figure's position so you can visually see its new placement compared to its original layout. For circles, these transformations make it clear how shifts affect their center locations.


When sketching the original circle equation \(x^2 + y^2 = 49\), you draw a circle centered at the origin with a radius extending 7 units in all directions. The graph portrays the perfect symmetry around the line intercepts.

• Sketching the shifted circle: Using the new equation \((x+2)^2 + (y+3)^2 = 49\), you then draw the transformed circle centered at \((-2,-3)\). This requires moving the circle without altering its size or circular shape.
• Labeling the graphs: It's crucial to have distinct labels for both graphs. This means clearly writing the equations for each circle near its representation. Labeling helps differentiate between the original and transformed shapes in any exercise or real-world application.

By completing these steps, you gain a visual understanding of the transformation, making it easier to grasp the impact of coordinate shifts on the graph's layout.