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. (x+1)(x+2)(x+3)≥0
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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 29
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)=x2+3x−10
Verified step by step guidance1
Identify the quadratic function given: \(f(x) = x^2 + 3x - 10\).
Find the vertex of the parabola using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = 1\) and \(b = 3\).
Calculate the y-coordinate of the vertex by substituting the x-value found into the function: \(f\left(-\frac{b}{2a}\right)\).
Find the x-intercepts by solving the quadratic equation \(x^2 + 3x - 10 = 0\) using factoring or the quadratic formula.
Determine the y-intercept by evaluating \(f(0)\), then write the equation of the axis of symmetry as \(x = -\frac{b}{2a}\), and use the vertex and intercepts to describe the domain and range of the function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex of a Quadratic Function
The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. It can be found using the formula x = -b/(2a) for a quadratic function f(x) = ax² + bx + c. The vertex helps in sketching the graph and determining the axis of symmetry.
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Vertex Form
Intercepts of a Quadratic Function
Intercepts are points where the graph crosses the axes. The y-intercept is found by evaluating f(0), and the x-intercepts (roots) are found by solving f(x) = 0. These points provide key reference locations for sketching the parabola accurately.
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Domain and Range of a Quadratic Function
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 values greater than or equal to the vertex's y-coordinate; if downward, all values less than or equal to it. This helps describe the function's output values.
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Related Practice
Textbook Question
Divide using synthetic division. (2x5−3x4+x3−x2+2x−1)/(x+2)
Textbook Question
In Exercises 27–29, divide using long division.
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. (x−1)(x−2)(x−3)≥0
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Textbook Question
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; i and 3i are zeros; f(-1) = 20
Textbook Question
Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero.
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