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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 5
Chapter 4, Problem 5

In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. f(x)=−x2+14x−106f(x) = -x^2 + 14x - 106

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1
Identify the type of function given. Since the function is \(f(x) = -x^2 + 14x - 106\), it is a quadratic function with a negative leading coefficient, which means its graph is a parabola opening downward and it has a maximum value.
Find the vertex of the parabola, since the vertex gives the maximum or minimum value of a quadratic function. Use the vertex formula for the x-coordinate: \(x = \frac{-b}{2a}\), where \(a = -1\) and \(b = 14\) from the function \(f(x) = ax^2 + bx + c\).
Calculate the y-coordinate of the vertex by substituting the x-value found into the original function: \(f(x) = -x^2 + 14x - 106\). This y-value is the maximum value of the function.
Determine the domain of the function. Since it is a quadratic function, the domain is all real numbers, which can be written as \((-\infty, \infty)\).
Determine the range of the function. Because the parabola opens downward and the vertex represents the maximum value, the range is all real numbers less than or equal to the maximum y-value found at the vertex. Express the range as \((-\infty, \text{maximum value}]\).

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Key Concepts

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

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 helps identify whether the function has a maximum or minimum value.
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Graphs of Logarithmic Functions

Vertex of a Parabola

The vertex of a parabola given by f(x) = ax^2 + bx + c is the point where the function attains its maximum or minimum value. It can be found using the formula x = -b/(2a). Substituting this x-value back into the function gives the corresponding y-value, which is the max or min.
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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 parabola's direction: if it opens downward, the range is all values less than or equal to the vertex's y-value; if upward, all values greater than or equal to it.
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Domain & Range of Transformed Functions
Related Practice
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−7)≤0

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Textbook Question

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

f(x)=x2+2x+1g(x)=x2−2x+1f(x) = x^2 + 2x + 1 \(\quad\) g(x) = x^2 - 2x + 1

h(x)=x2−1j(x)=−x2−1h(x) = x^2 - 1 \(\quad\) j(x) = -x^2 - 1

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. h(x)=7x3+2x2+1xh(x)=7x^3+2x^2+\(\frac{1}{x}\)

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Textbook Question

Divide using long division. State the quotient, and the remainder, r(x). (x3+5x2+7x+2)÷(x+2)

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x7+πx5+2/3 x