Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Match each equation with the appropriate description . Do not use a calculator. $$(x-3)^{2}+(y+4)^{2}=25$$ A. Circle; center \((3,-4) ;\) radius 5 B. Parabola; opens left C. Parabola; opens upward D. Circle; center \((-3,4)\); radius 5 E. Parabola; opens right F. Circle; center \((0,0) ;\) radius \(\sqrt{5}\) G. No points on its graph H. Parabola; opens downward

Step-by-Step Solution

Answer

A. Circle; center (3,-4); radius 5

Step 1: Identify the Equation Type

The given equation \((x-3)^{2}+(y+4)^{2}=25\) is in the form of \((x-h)^2 + (y-k)^2 = r^2\), which characterizes the equation of a circle in the coordinate plane.

Step 2: Identify the Center of the Circle

The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle. Comparing \((x-3)^2 + (y+4)^2 = 25\) with the standard form, we find that \((h, k) = (3, -4)\).

Step 3: Identify the Radius of the Circle

In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the \(r^2\) term is the square of the radius. In our equation, \(r^2 = 25\). Thus, the radius \(r = \sqrt{25} = 5\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric shapes. It helps us understand and describe the relationships between points, lines, and shapes on a coordinate plane.

In coordinate geometry, each point is represented by a pair of numbers, often designated as \(x,y\), where \(x\) indicates the horizontal position and \(y\) the vertical position. By using equations, we can depict a variety of geometric entities, such as lines, curves, and circles, within this plane.

For instance, the equation \(x^2 + y^2 = r^2\) represents a circle centered at the origin with a radius \(r\). Adjusting the equation to \((x-h)^2 + (y-k)^2 = r^2\) shifts the center of the circle to \(h, k\), which allows for even more flexibility in describing the location and size of circles and other shapes. This mathematical language provides us a powerful tool to precisely represent and study these shapes.

Equation Matching

Equation matching is a critical skill in coordinate geometry, allowing us to interpret and match mathematical equations to corresponding geometric descriptions.

In our example, the equation \( (x-3)^2 + (y+4)^2 = 25 \) is in the standard form of a circle, which is \( (x-h)^2 + (y-k)^2 = r^2 \). When matching equations, it is essential to first recognize the type of geometric shape the equation represents.

Upon identifying the type, proceed to extract characteristics such as the center and radius for circles, or the vertex and direction of opening for parabolas. Consider:

• Look at the equation's components to identify whether it's a circle, parabola, or any other shape.
• Use clues from the equation form to find specific parameters like centers or orientations.

By accurately matching equations, we can correctly pair them with their geometric descriptions, enhancing our understanding of the relationship between algebraic expressions and geometric forms.

Graphical Interpretation

Graphical interpretation is the process of visualizing equations as shapes or paths on the coordinate plane, providing a bridge between abstract equations and tangible visual forms.

To interpret the equation \( (x-3)^2 + (y+4)^2 = 25 \) graphically, we first identify it as a circle with a center at \( (3, -4) \) and a radius of \( 5 \).

Visualizing this:

• The center point \( (3, -4) \) is where the circle is centered in the coordinate plane.
• From this center, measure a distance equal to the radius \( r = 5 \) in all directions to draw the circle.

Graphical interpretation not only aids in comprehending the shape and position of geometric figures but also helps in solving problems related to intersections, tangents, and areas within the geometry. By practicing how to graphically interpret equations, students can strengthen their spatial reasoning and connect theoretical math with practical visual outcomes.