Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given equation. $$ (x+3)^{2}+(y-4)^{2}=25 $$

Step-by-Step Solution

Answer

The graph is a circle centered at (-3, 4) with a radius of 5.

Step 1: Identify the Type of Equation

The given equation is \((x+3)^2 + (y-4)^2 = 25\). This is the equation of a circle in the standard form \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Determine the Center of the Circle

From the equation \((x+3)^2 + (y-4)^2 = 25\), we can identify the center \((h, k)\) as \((-3, 4)\).

Step 3: Identify the Radius

The right-hand side of the equation is 25, which is the square of the radius \(r\). Thus, the radius \(r\) is \(\sqrt{25} = 5\).

Step 4: Sketch the Graph

Draw a coordinate system. Mark the center of the circle at the point \((-3, 4)\). From this center, measure the radius of 5 units in all directions (up, down, left, right) to outline the circle.

Key Concepts

Graph Sketching

Graph sketching is a visual representation of equations and functions on a coordinate plane. It helps us better understand the relationship between variables. When sketching the graph of a circle equation, it's essential to identify the key elements: center, radius, and overall shape. First, draw the coordinate axes with clear labels. Then, locate the center of the circle using its coordinates and mark this point on the graph. From the center, use the radius to mark points at equal distances in all directions: up, down, left, and right. Connect these points smoothly to form a circle. Remember, the focus is on accuracy and symmetry for a well-drawn circle. Keeping the graph neat helps in visualizing and understanding the equation's solution.

Equation of a Circle

The equation of a circle is a way to describe a circle on a coordinate plane. The standard form of a circle equation is \[(x-h)^2 + (y-k)^2 = r^2\]where

• \( (h, k) \) is the center of the circle, a setpoint in the coordinate plane from which every point on the circle is equidistant.
• \( r \) is the radius of the circle, indicating the distance from the center to any point on the circle.

Understanding the equation allows you to pinpoint the circle's location and size in the graph. For instance, in the equation \((x+3)^2 + (y-4)^2 = 25\),the center is \((-3, 4)\)and \(r^2 = 25\),leading to \(r = 5\). This clarity in visualization aids in graph sketching and provides an accurate representation of circles in coordinate geometry.

Radius and Center Calculation

To sketch a circle, calculating the center and radius from its equation is fundamental. Let's break this down using the given equation \((x+3)^2 + (y-4)^2 = 25\). ### Calculating the CenterThe center of a circle in its equation form \((x-h)^2 + (y-k)^2 = r^2\) is \((h, k)\). Compare this with \((x+3)^2 + (y-4)^2\),and you'll find \(h = -3\)and \(k = 4\). So, the center is \((-3, 4)\).### Determining the RadiusOn the right side of the equation, \(r^2 = 25\).Taking the square root, we find that \(r = \sqrt{25} = 5\).This straightforward calculation enables us to visualize the circle accurately on the graph. With the center and radius known, it becomes easier to sketch and analyze the circle effectively.