Domanda del libro di testo
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 60
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 - 6y -7=0
Guida verificata passo dopo passo1
Start with the given equation: \(x^{2} + y^{2} - 6y - 7 = 0\).
Group the \(y\) terms together and isolate the constant on the other side: \(x^{2} + (y^{2} - 6y) = 7\).
Complete the square for the \(y\) terms. Take half of the coefficient of \(y\), which is \(-6\), divide by 2 to get \(-3\), then square it to get \(9\). Add \(9\) to both sides to keep the equation balanced: \(x^{2} + (y^{2} - 6y + 9) = 7 + 9\).
Rewrite the perfect square trinomial as a binomial squared: \(x^{2} + (y - 3)^{2} = 16\).
Identify the center and radius of the circle from the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, the center is \((0, 3)\) and the radius is \(\sqrt{16}\).
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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 a constant. This technique helps transform the equation of a circle into its standard form, making it easier to identify key features like the center and radius.
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06:24
Solving Quadratic Equations by Completing the Square
Standard Form of a Circle
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Writing the equation in this form allows direct identification of the circle's center coordinates and radius, which are essential for graphing.
Video consigliato:
5:18Circles in Standard Form
Graphing Circles
Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points equidistant from the center. Understanding how to interpret the standard form equation enables accurate sketching of the circle's shape and position.
Video consigliato:
5:18Circles in Standard Form
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Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. ƒ-1 (1)
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In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2
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Domanda del libro di testo
In Exercises 59-66, a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. 4x+y-6=0
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