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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 39
Chapter 4, Problem 39

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=3x2−12x−1

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Identify the coefficient of the quadratic term in the function \(f(x) = 3x^2 - 12x - 1\). Here, the coefficient \(a = 3\). Since \(a > 0\), the parabola opens upward, which means the function has a minimum value.
Find the vertex of the parabola, which gives the minimum value and where it occurs. Use the vertex formula for the \(x\)-coordinate: \(x = -\frac{b}{2a}\). Here, \(b = -12\) and \(a = 3\), so calculate \(x = -\frac{-12}{2 \times 3}\).
Substitute the \(x\)-value of the vertex back into the function to find the minimum value: \(f\left(-\frac{b}{2a}\right) = 3\left(-\frac{b}{2a}\right)^2 - 12\left(-\frac{b}{2a}\right) - 1\).
Determine the domain of the function. Since \(f(x)\) is a quadratic function, its domain is all real numbers, which can be written as \((-\infty, \infty)\).
Determine the range of the function. Since the parabola opens upward and has a minimum value at the vertex, the range is all real numbers greater than or equal to the minimum value found in step 3. Express the range as \([\text{minimum value}, \infty)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. This determines whether the function has a minimum (a > 0) or maximum (a < 0) value.
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Properties of Parabolas

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its minimum or maximum value. It can be found using the formula x = -b/(2a). Substituting this x-value back into the function gives the corresponding minimum or maximum value.
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Horizontal Parabolas

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 and the direction the parabola opens: if it opens upward, the range is from the minimum value to infinity; if downward, from negative infinity to the maximum value.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x3−4x2−7x+10

Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−8x2+x+2; between 2 and 3

Textbook Question

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x4+10x3+5x2+x+1; f(−2/3)

Textbook Question

Use Descartes' Rule of Signs to explain why 2x4+6x2+8=02x^4 + 6x^2 + 8 = 0 has no real roots.

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. x3+x2+4x+4>0x^3+x^2+4x+4>0

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Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x2/(3x2+1)