Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 104

Chapter 3, Problem 104

Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

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, \(h = 1\), \(k = -1\), and \(r = 1\).

Substitute these values into the standard form equation to write the specific equation of the circle: \[ (x - 1)^2 + (y + 1)^2 = 1^2 \]

To graph the circle, first plot the center point at \((1, -1)\) on the rectangular coordinate system.

From the center, use the radius to mark points 1 unit away in all four directions (up, down, left, right), then sketch the circle passing through these points.

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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 with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². This formula represents all points (x, y) that are exactly r units away from the center. Understanding this equation is essential for graphing the circle accurately.

Coordinate Plane and Plotting Points

A rectangular coordinate system consists of an x-axis and y-axis intersecting at the origin. Plotting points involves locating coordinates (x, y) on this plane. Knowing how to plot the center and points on the circle helps visualize and draw the circle correctly.

Radius and Distance in the Plane

The radius is the fixed distance from the center of the circle to any point on its circumference. Understanding how to measure and use this distance in the coordinate plane allows you to determine points on the circle by moving r units horizontally or vertically from the center.

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Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. r(x) = (x − 2)³ +1

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Solve and graph the solution set on a number line: 3|2x-1| ≥ 21

Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.

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Textbook Question

Begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = x³/2

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Textbook Question

In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)

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Exercises 103–105 will help you prepare for the material covered in the next section. Solve by completing the square: y² – 6y — 4 = 0.

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