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Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Informally assess the fit of a function by plotting and analyzing residuals.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Determine an explicit expression, a recursive process, or steps for calculation from a context. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Interpret the parameters in a linear or exponential function in terms of a context.