Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 55

Chapter 3, Problem 55

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² – 10x – 6y – 30 = 0

Verified step by step guidance

Start with the given equation: \(x^{2} + y^{2} - 10x - 6y - 30 = 0\).

Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(\left(x^{2} - 10x\right) + \left(y^{2} - 6y\right) = 30\).

Complete the square for the \(x\) terms: take half of \(-10\), which is \(-5\), then square it to get \(25\). Add \(25\) inside the \(x\) group.

Complete the square for the \(y\) terms: take half of \(-6\), which is \(-3\), then square it to get \(9\). Add \(9\) inside the \(y\) group.

Since you added \(25\) and \(9\) to the left side, add the same amounts to the right side to keep the equation balanced: \(30 + 25 + 9\).

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Key Concepts

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

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting appropriate constants. This technique helps transform the general form of a circle's equation into its standard form, making it easier to identify key features like the center and radius.

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Writing the equation in this form allows for straightforward identification of the circle's geometric properties and simplifies graphing.

Graphing Circles

Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points at a distance r in all directions. Understanding how to interpret the standard form equation aids in accurately sketching the circle and visualizing its position and size.

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Circles in Standard Form

Related Practice

Textbook Question

Find the average rate of change of f(x) = x^2 - 4x from x_1 = 5 to x_2 = 9.

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Textbook Question

In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function. (go f) (-1)

Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

Textbook Question

Find

a. (fog) (x)

b. (go f) (x)

c. (fog) (2)

d. (go f) (2).

f(x) = x+4, g(x) = 2x + 1

Textbook Question

In Exercises 55–59, use the graph of to graph each function g.

g(x) = f(x + 2) + 3

Textbook Question

In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = √x (x = 0, 1, 4, 9) and g (x) = √(x + 2) (x = = −2, −1, 2, 7)