Domanda del libro di testo
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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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 4, Problema 13
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x2−8x+3
Guida verificata passo dopo passo1
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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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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Solving Quadratic Equations Using The Quadratic Formula
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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5:28Horizontal Parabolas
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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06:24Solving Quadratic Equations by Completing the Square
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
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Domanda del libro di testo
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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Domanda del libro di testo
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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