Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 55

3장, 문제 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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

관련 실천

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

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교과서 질문

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

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교과서 질문

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

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교과서 질문

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

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

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교과서 질문

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)

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교과서 질문

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (x − 2)²

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