Define appropriate quantities for the purpose of descriptive modeling.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a) Factor a quadratic expression to reveal the zeros of the function it defines. b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c) Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be written as {(1.15^(1/12)}^12t~1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
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
Solve quadratic equations in one variable. a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Derive the quadratic formula from this form. b) Solve quadratic equations by inspection (e.g., for x^2 =49), taking the square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b.
Choose a level of accuracy appropriate to limitations on measurement when reporting.
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
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. EX. 5 ^ (1/3) to be the cube root of 5 because we want {5^(1/3)}^3 = 5^(1/3)*3 to hold, so {5(1/3)}^3 must equal 5.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Find inverse functions. A) Solve an equation of form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. B) (+) Verify by composition that one function is the inverse of another. C) (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. D) (+) Produce an invertible function from a non-invertible function by restricting the domain.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.