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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 41
Chapter 4, Problem 41

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)=−4x2+8x−3

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1
Identify the coefficient of the quadratic term in the function \(f(x) = -4x^2 + 8x - 3\). Here, the coefficient \(a = -4\).
Determine whether the function has a minimum or maximum value by looking at the sign of \(a\). Since \(a = -4\) is negative, the parabola opens downward, so the function has a maximum value.
Find the vertex of the parabola, which gives the maximum value and where it occurs. Use the vertex formula for the \(x\)-coordinate: \(x = \frac{-b}{2a}\), where \(b = 8\) and \(a = -4\).
Calculate the \(y\)-coordinate of the vertex by substituting the \(x\)-value back into the function: \(f\left(\frac{-b}{2a}\right) = -4\left(\frac{-b}{2a}\right)^2 + 8\left(\frac{-b}{2a}\right) - 3\).
Identify the domain and range of the function. The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range depends on the maximum value found at the vertex; since the parabola opens downward, the range is \((-\infty, \text{maximum value}]\).

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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 all values greater than or equal to the minimum; if downward, all values less than or equal to the maximum.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

In Exercises 39–52, 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. 2x3−x2−9x−4=0

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Textbook Question

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. f(x)=x3+3x2−4f(x) = x^3 + 3x^2 - 4

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≥9x2x^3≥9x^2

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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. x3≤4x2

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Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x3/(3x2+1)

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Textbook Question

Use synthetic division to divide f(x)=x3−4x2+x+6 by x+1. Use the result to find all zeros of f.