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. 6x2+x>1
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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
In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x3+x2−3x+1
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Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As _____
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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.
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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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