Textbook Question
In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x4+6x3−18x2; between 2 and 3
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 37
In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x2−2
Verified step by step guidance1
Rewrite the function in standard quadratic form: \(f(x) = -x^{2} + 2x - 2\).
Find the vertex using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = -1\) and \(b = 2\).
Calculate the y-coordinate of the vertex by substituting the x-value back into the function: \(f(x) = -x^{2} + 2x - 2\).
Find the y-intercept by evaluating \(f(0)\), and find the x-intercepts by solving the equation \(-x^{2} + 2x - 2 = 0\).
Write the equation of the axis of symmetry as \(x = \) (the x-coordinate of the vertex), then determine the domain (all real numbers) and the range based on the vertex and the direction the parabola opens.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Functions and Their Graphs
A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and key features of the parabola helps in sketching the graph accurately.
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Graphs of Logarithmic Functions
Vertex and Axis of Symmetry
The vertex is the highest or lowest point on the parabola, found using the formula x = -b/(2a). The axis of symmetry is a vertical line through the vertex that divides the parabola into two mirror images. Knowing the vertex and axis helps in plotting the graph and understanding its symmetry.
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Domain and Range of Quadratic Functions
The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward, the range is all y-values greater than or equal to the vertex's y-coordinate; if downward, all y-values less than or equal to the vertex's y-coordinate. This helps describe the function's output values.
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Related Practice
Textbook Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=2x4−5x3−x2−6x+4
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Textbook Question
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x3+x2−2x+1; between -3 and -2
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.
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Textbook Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x2+1)
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Textbook Question
Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function.