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Explore the concepts of power and root functions, including their definitions, graphs, properties, and applications in modeling and graphing. Learn how to transform and graph power and root functions and apply them to solve equations and inequalities. Understand the domain restrictions and apply these functions to model real-life scenarios.
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Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Rational Functions and Graphs 4.3 Rational Equations, Inequalities, Models, and Applications 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions
4.4 Functions Defined by Powers and Roots Power and Root Functions A function f given by f(x) = xb, where b is a constant, is a power function. If , for some integer n 2, then f is a root function given by f(x) = x1/n, or equivalently, f(x) = • f(x) = xp/q, p/q in lowest terms • if q is odd, the domain is all real numbers • if q is even, the domain is all nonnegative real numbers
4.4 Graphing Power Functions Example Graph f(x) = xb, b = 0.3, 1, and 1.7, for x 0. Solution The larger values of b cause the graph of f to increase faster.
4.4 Applying Properties of Rational Exponents Example Simplify each expression by hand. (a) (b) Solution (a) (b)
4.4 Modeling Wing Size of a Bird Example Heavier birds have larger wings with more surface area. For some species of birds, this relationship can be modeled by S (x) = 0.2x2/3, where x is the weight of the bird in kilograms and S is the surface area of the wings in square meters. Approximate S(0.5) and interpret the result. Solution The wings of a bird that weighs 0.5 kilogram have a surface area of about 0.126 square meter.
4.4 Modeling the Length of a Bird’s Wing Example The table lists the weight W and the wingspan L for birds of a particular species. • Use power regression to model the data with L = aWb. Graph the data and the equation. (b) Approximate the wingspan for a bird weighing 3.2 kilograms. W (in kilograms) L (in meters)
4.4 Modeling the Length of a Bird’s Wing Solution (a) Let x be the weight W and y be the length L. Enter the data, and then select power regression (PwrReg), as shown in the following figures.
4.4 Modeling the Length of a Bird’s Wing The resulting equation and graph can be seen in the figures below. (b) If a bird weighs 3.2 kg, this model predicts the wingspan to be
4.4 Finding Domains of Root Functions Example Find the domain of each function. (a) (b) Solution • 4x + 12 must be greater than or equal to 0 since the root, n = 2, is even. (b) Since the root, n = 3, is odd, the domain of g is all real numbers. The domain of f is [–3,).
4.4 Transforming Graphs of Root Functions Example Explain how the graph of can be obtained from the graph of Solution Shift left 3 units and stretch vertically by a factor of 2.
4.4 Transforming Graphs of Root Functions Example Explain how the graph of can be obtained from the graph of Solution Shift right 1 unit, stretch vertically by a factor of 2, and reflect across the x-axis.
4.4 Graphing Circles Using Root Functions • The equation of a circle centered at the origin with radius r is foundby finding the distance from the origin to a point (x,y) on the circle. • The circle is not a function, so imagine a semicircle on top and another on the bottom.
4.4 Graphing Circles Using Root Functions • Solve for y: • Since y2 = –y1, the “bottom” semicircle is a reflection of the “top” semicircle.
4.4 Graphing a Circle Example Use a calculator in function mode to graph the circle Solution This graph can be obtained by graphing in the same window. Technology Note: Graphs may not connect when using a non-decimal window.
