General, Recurring Skills
Algebraically Solve Equations · Given an equation consisting of a composite of linear, quadratic, simple algebraic, exponential, or logarithmic functions of a variable, solve for that variable.
Translate Between Algebraic and Geometric Perspectives · Given an algebraic, exponential, or logarithmic function \(f,\) know how the algebraic features of the formula \(f(x)\) correspond to the geometric features of the graph \(y = f(x).\)
Mathematically Model Phenomena · Given two-variable numerical data \(\big\{(x_i, y_i)\big\}\) for a phenomenon, decide what type of function \(f\) appropriately models the phenomenon, use technology to perform regression to determine the model \(f(x)\) of that type of function that best fits the data, and interpret the algebraic and geometric features of the model in the context of the phenomenon.
Specific Skills
- Use technology to determine a decimal approximation of a number expressed in terms of algebraic, exponential, or logarithmic operations.
- Infer from the formula for a function the largest subset of the real numbers that could serve as the function’s domain.
- Given a formula and domain of a function, determine its range.
- Infer the domain and range of a function from its graph.
- Given the coordinates of two points in the \(xy\)-plane, determine an equation of the line that passes through them.
- Given the equations of two non-parallel lines in the \(xy\)-plane, determine the coordinates of the point at which they intersect. I.e. solve a system of two linear equations.
- Given the equation of a line and coordinates of a point in the \(xy\)-plane, determine an equation of the line passing through that point that is either parallel or perpendicular to that line.
- Given a quadratic polynomial function \(f\) for which \(f(x)\) is expressed in one of the following forms, rewrite it in either of the other forms. \[ ax^2+bx+c \qquad a(x-h)^2+k \qquad a(x-r_1)(x-r_2) \]
- Sketch the graph of a piecewise-defined function.
- For an account with initial balance \(P\) that appreciates to a balance of \(S\) after \(t\) years earning interest an at annual interest rate \(r\) compounded \(n\) times per year, Given a value for \(n\) and given values for any three of the parameters \(P\), \(S\), \(t,\) and \(r,\) solve for the other parameter.
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For an account with initial balance zero that appreciates to a balance of \(S\) after \(t\) years earning interest an at annual interest rate \(r\) compounded \(n\) times per year along with a regular deposit of \(P\) at the time of the interest payment, given values of \(n\) and \(r\) and values of any two of the parameters \(P,\) \(S,\) or \(t,\) solve for the other parameter.
And similarly solve for a parameter if instead regular withdrawals of \(P\) are made from an account with initial balance \(S\) that depreciates to a balance of zero after \(t\) years.
- Identify the rational roots of a polynomial.
- Given a polynomial \(f\) with root \(r,\) calculate the quotient polynomial \(q\) such that \( f(x) = (x-r)\times q(x)\,. \)
- Sketch the graph of a polynomial or rational function presented in factored form.
- Given the graph \(y = f(x)\) of a function \(f,\) sketch the graph of the linearly transformed function \(y = af(cx+d)+b.\)
- Given formulas \(f(x)\) and \(g(x)\) for functions \(f\) and \(g\) write down formulas for the functions \(f \pm g,\) \(f \times g,\) \(f/g,\) and \(f \circ g\,.\)
- Given an invertible function \(f\) defined by a formula \(f(x)\) featuring a single instance of its independent variable, write a formula \(f^{-1}(x)\) for the inverse of that function.