Domanda del libro di testo
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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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 61
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 − x + 2y + 1 = 0
Guida verificata passo dopo passo1
Start with the given equation: \(x^{2} + y^{2} - x + 2y + 1 = 0\).
Group the \(x\) terms and \(y\) terms together: \((x^{2} - x) + (y^{2} + 2y) = -1\) (move the constant term to the right side).
Complete the square for the \(x\) terms: take half of the coefficient of \(x\) (which is \(-1\)), square it, and add inside the parentheses. Half of \(-1\) is \(-\frac{1}{2}\), and its square is \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). Do the same for the \(y\) terms: half of \(2\) is \(1\), and its square is \(1^2 = 1\).
Add these squares to both sides of the equation to keep it balanced: \((x^{2} - x + \frac{1}{4}) + (y^{2} + 2y + 1) = -1 + \frac{1}{4} + 1\).
Rewrite each perfect square trinomial as a squared binomial: \((x - \frac{1}{2})^{2} + (y + 1)^{2} = \text{(simplify the right side)}\). This is the standard form of a circle equation, where the center is at \(\left(\frac{1}{2}, -1\right)\) and the radius is the square root of the right side.
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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 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.
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06:24
Solving Quadratic Equations by Completing the Square
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.
Video consigliato:
5:18Circles in Standard Form
Identifying the Center and Radius from the Equation
Once the equation is in standard form, the center of the circle is given by the coordinates (h, k), and the radius is the square root of the constant on the right side. Understanding how to extract these values is essential for graphing the circle accurately and interpreting its position and size.
Video consigliato:
5:18Circles in Standard Form
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