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College algebra concepts and contexts8/23/2023 Graphs of some power functions are shown.ĪBOUT THE AUTHORS JAMES STEWART received his MS from Stanford University and his PhD from the University of Toronto. yĪ power function is a function of the form f 1x 2 = Cx p. The vertex of the graph of f is at the point (h, k). The function f can be expressed in the standard form f 1x2 = a1x - h2 2 + k. The maximum or minimum value of f occurs at x =. The graph of f has the shape of a parabola. The graph of f has the general shape shown below.Ī quadratic function is a function of the form f 1x2 = ax 2 + bx + c. yĪ logarithmic function with base a 7 1 is a function of the form f 1x2 = loga x. If a 6 1, then a is called the decay factor and r = a - 1 is called the decay rate. If a 7 1, then a is called the growth factor and r = a - 1 is called the growth rate. The graph of f has one of the shapes shown. yĪn exponential function is a function of the form f 1x2 = Ca x. The graph of f is a line with slope m and y-intercept b. A linear function is a function of the form f 1x2 = b + mx. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the ![]() ![]() The properties of exponents apply to rational exponents.Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals.These roots have the same properties as square roots. The principal nth root of a a is the number with the same sign as a a that when raised to the nth power equals a.To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. Radical expressions written in simplest form do not contain a radical in the denominator.We can add and subtract radical expressions if they have the same radicand and the same index.If a a and b b are nonnegative, the square root of the quotient a b a b is equal to the quotient of the square roots of a a and b b See Example 4 and Example 5.If a a and b b are nonnegative, the square root of the product a b a b is equal to the product of the square roots of a a and b b See Example 2 and Example 3.The principal square root of a number a a is the nonnegative number that when multiplied by itself equals a.Scientific notation may be used to simplify calculations with very large or very small numbers.Scientific notation uses powers of 10 to simplify very large or very small numbers.The rules for exponential expressions can be combined to simplify more complicated expressions.The power of a quotient of factors is the same as the quotient of the powers of the same factors.The power of a product of factors is the same as the product of the powers of the same factors.An expression with a negative exponent is defined as a reciprocal.An expression with exponent zero is defined as 1.Powers of exponential expressions with the same base can be simplified by multiplying exponents.Quotients of exponential expressions with the same base can be simplified by subtracting exponents.Products of exponential expressions with the same base can be simplified by adding exponents.They may be simplified or evaluated as any mathematical expression. Formulas are equations in which one quantity is represented in terms of other quantities.See Example 9, Example 10, and Example 12 They take on a numerical value when evaluated by replacing variables with constants. Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division.These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. ![]()
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