Textbook Question
Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).
f(x) = 1/x, g(x)= 1/x
Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 3, Problem 64
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+3x+5y+9/4=0
Verified step by step guidance1
Start with the given equation: \(x^{2} + y^{2} + 3x + 5y + \frac{9}{4} = 0\).
Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(x^{2} + 3x + y^{2} + 5y = -\frac{9}{4}\).
Complete the square for the \(x\) terms: take half of the coefficient of \(x\) (which is \(3\)), square it, and add it inside the equation. Half of \(3\) is \(\frac{3}{2}\), and its square is \(\left(\frac{3}{2}\right)^{2} = \frac{9}{4}\).
Complete the square for the \(y\) terms: take half of the coefficient of \(y\) (which is \(5\)), square it, and add it inside the equation. Half of \(5\) is \(\frac{5}{2}\), and its square is \(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\).
Add these squares to both sides of the equation to keep it balanced: \(x^{2} + 3x + \frac{9}{4} + y^{2} + 5y + \frac{25}{4} = -\frac{9}{4} + \frac{9}{4} + \frac{25}{4}\), then rewrite the left side as perfect square trinomials and simplify the right side.
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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 + p)² = q. It involves adding and subtracting a constant to create a perfect square trinomial, which simplifies solving or rewriting equations, especially for conic sections like circles.
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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. Converting the general form to this form helps identify the circle's key features and makes graphing straightforward.
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5:18Circles in Standard Form
Identifying the Center and Radius
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. This information is essential for graphing and understanding the circle's position and size.
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05:01Identifying Intervals of Unknown Behavior
Related Practice
Textbook Question
Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = (1/2) (x − 1)² – 1
Textbook Question
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. f(g[h (1)])
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Textbook Question
In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x + 3)
Textbook Question
Use the graph of f to find each indicated function value.
f(-2)
Textbook Question
Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).
f(x) = 2x-3, g(x) = (x+3)/2