Textbook Question
Identify which graphs are not those of polynomial functions.
Ch. 3 - Polynomial and Rational Functions
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 4, Problem 13
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x2−8x+3
Verified step by step guidance1
Identify the quadratic function given: \(f(x) = 2x^2 - 8x + 3\).
Recall that the vertex of a parabola defined by \(f(x) = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\).
Substitute the values of \(a = 2\) and \(b = -8\) into the formula: \(x = -\frac{-8}{2 \times 2}\).
Simplify the expression to find the x-coordinate of the vertex.
To find the y-coordinate of the vertex, substitute the x-coordinate back into the original function: \(f(x) = 2x^2 - 8x + 3\), and simplify.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Functions
A quadratic function is a polynomial of degree two, generally written as f(x) = ax² + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient a.
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Vertex of a Parabola
The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For f(x) = ax² + bx + c, the vertex's x-coordinate is found using -b/(2a), and the y-coordinate is f(-b/(2a)).
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Completing the Square and Vertex Formula
Completing the square is a method to rewrite a quadratic function in vertex form, revealing the vertex coordinates directly. Alternatively, the vertex formula uses coefficients a and b to find the vertex without rewriting the function.
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Related Practice
Textbook Question
In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x3+13x2−11x−15)/(3x2−x−3)
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Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (x4−81)/(x−3)
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube of z and inversely as y.
Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x2+3x−2≥0
Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 2x2+x<15
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