Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 2

Chapter 3, Problem 2

Fill in the blank(s) to correctly complete each sentence. The circle with center (3, 6) and radius 4 has equation _________.

Verified step by step guidance

Recall the standard form of the equation of a circle with center $(h, k)$ and radius $r$ is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Identify the center $(h, k)$ and radius $r$ from the problem: here, the center is $(3, 6)$ and the radius is $4$.

Substitute $h = 3$, $k = 6$, and $r = 4$ into the standard form equation: \[ (x - 3)^2 + (y - 6)^2 = 4^2 \]

Simplify the right side by squaring the radius: \[ (x - 3)^2 + (y - 6)^2 = 16 \]

Write the final equation of the circle as: \[ (x - 3)^2 + (y - 6)^2 = 16 \]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle

The equation of a circle in the coordinate plane is given by (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This formula represents all points (x, y) that are exactly r units away from the center.

Coordinates of the Center

The center of a circle is a fixed point from which every point on the circle is equidistant. In the equation, the center coordinates (h, k) shift the circle from the origin to the point (3, 6) in this problem.

Radius and Its Role in the Equation

The radius is the distance from the center to any point on the circle. Squaring the radius (r²) in the equation ensures the distance formula is correctly represented, so for radius 4, r² equals 16.

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Fill in the blank to correctly complete each sentence. The point (4,_____ ) lies on the graph of the equation y = 3x - 6.

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To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x³? What is its range?

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