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Pre-Calculus Unit 1 – Functions and Their Graphs Scoring Criteria Performance Indicators U1.PI 1 Use multiple representations to describe linear functional relationships. U1.PI 2 Graph piecewise functions. “1” – Emerging I have demonstrated the most basic knowledge/skills relevant to this standard. “2” – Developing I have demonstrated relevant knowledge/skills, but have not yet demonstrated convincing evidence of fully meeting the standard. Level 1 I can calculate slope using two points on a line. I can identify and justify the slope of a vertical line (undefined) and horizontal line. L2: Level 1, plus… I can represent linear functions using slopeintercept form, graph a line given slope-intercept form and identify the slope and x and y-intercepts of a line. I can accurately plot the endpoints of all segments of a piecewise function. I can graph a linear piecewise function. CCSS Domain(s): Functions “3” – Achieving I have demonstrated that I have the knowledge/skills defined in the standard. “4” – Excelling I have demonstrated the knowledge/skills defined by the standard with a high level of understanding/ability as defined by the discipline. L3: Level 2, plus… I can demonstrate algebraic and graphical equivalence of linear functions using point-slope, slope-intercept and standard forms. L4: Level 3, plus… I can graph a piecewise function including linear and non-linear segments by hand and using graphing technology. I can determine the slope of a line using the difference quotient and explain how to derive the formula using function notation. U1.PI 3 Identify the domain and range of a function given an algebraic expression. Identify and verify inverse functions. I can identify the domain or range of a single function family from an algebraic expression. I can find an inverse of a given function given in a table, equation or graph and I can write the inverse function using appropriate notation. I can identify the domain and range of a single function family from an algebraic expression. I can create inverse functions algebraically and use composition of functions to verify the inverse relationship. I can create figures using transformations of multiple parent functions and domain restrictions. I can write a piecewise function for a given graph. I can identify the domain of a complex function from an algebraic expression. I can identify the domain and range of a complex function from an algebraic expression. I can explain if a function is invertible using domain/range and one-to-one terminology. I can produce an invertible function from a non-invertible function by restricting the domain. I can demonstrate an inverse function graphically. Pre-Calculus Unit 2 – Polynomial Functions and Rational Functions “1” – Emerging I have demonstrated the most basic knowledge/skills relevant to this standard. Scoring Criteria Performance Indicators U2PI4 Graphing and interpreting quadratic functions using key characteristics. U2PI5 Graphing higher order polynomials using the end behavior and multiplicity of zeroes. Identifying real (rational and irrational) zeros of higher order polynomials. “2” – Developing I have demonstrated relevant knowledge/skills, but have not yet demonstrated convincing evidence of fully meeting the standard. Level 1, plus… Level 2, plus… U2PI7 Graphing rational functions using key function characteristics. “4” – Excelling I have demonstrated the knowledge/skills defined by the standard with a high level of understanding/ability as defined by the discipline. Level 3, plus… I can identify the zeros of a quadratic function in standard form by using the quadratic formula or factoring. I can find an equation in standard form given the vertex and a point on the parabola or given the zeros and a point on the parabola. I can identify the vertex and zeros of a standard form quadratic function by completing the square when 𝑎 ≠ 1. I can identify key attributes of a quadratic function (vertex, axis of symmetry, zeros, y-intercept) from a graph. I can identify the vertex and zeros of a standard form quadratic function by completing the square when 𝑎 = 1. I can graph quadratic functions from standard, factored or vertex form. I can use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions and accurately describe the end behavior using limit notation. I can identify a polynomial equation in completely factored form. I can use transformations multiplicities of zeroes to sketch graphs of polynomial functions. I can identify the correct y-intercept and the leading coefficient “a” when graphing polynomial functions or when writing equations for polynomials from graphs. I can identify the zeros of a polynomial function using long or synthetic division. I can apply the Remainder and Factor Theorems. I can use the Rational Zero Test to determine possible rational zeros of polynomial functions. U2PI6 Graphing polynomials with real and complex zeros (Fundamental Theorem of Algebra). “3” – Achieving I have demonstrated that I have the knowledge/skills defined in the standard. I can use the imaginary unit “I” to write complex numbers. I can use commutative, associative and distributive properties to add, subtract and multiply complex numbers. I can identify vertical and horizontal asymptotes of rational functions. I can use complex conjugates to write the quotient of two complex numbers in standard form. I can use the Intermediate Value Theorem to help locate zeros of polynomial functions. I can use Descartes’ Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. I can apply the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function. I can find all zeros of polynomial functions including complex zeros. I can create a sketch of a rational function involving vertical and horizontal asymptotes. I can identify hole in the graph of a rational function. I can create a sketch of a rational function with two vertical asymptotes or a slant asymptote showing the end behavior. I can solve real life situation problems using key attributes of quadratic functions. I can identify rational and irrational zeroes of a polynomial equation given in factored form and sketch an accurate graph involving irrational zeroes. I can find the real zeros of a polynomial algebraically and use the zeros, end behavior and additional points to sketch a graph of the function. I can find all zeros of any higher order polynomial (real and complex) by factoring and can sketch the resulting graph. I can graph rational functions and accurately identify zeroes, asymptotes, y-intercepts and additional points as needed. CCSS Domains and Standards: NUMBER AND QUANTITY: The Complex Number System N-CN 1. Know there is a complex number 𝑖 such that 𝑖² = −1, and every complex number has the form 𝑎 + 𝑏𝑖 with 𝑎 and 𝑏 real. N-CN 2. Use the relation 𝑖² = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N-CN 3. Find the conjugate of a complex number, use conjugates to find moduli and quotients of complex numbers. Use complex numbers in polynomial identities and equations. N-CN 7. Solve quadratic equations with real coefficients that have complex solutions N-CN 8. Extend polynomials identities to the complex number. For example, rewrite 𝑥² + 4 = (𝑥 + 2𝑖)(𝑥 − 2𝑖) N-CN 9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. ALGEBRA: Interpret Structure of expressions A-SSE 2. Use the structure of expression to identify ways to rewrite it. For example, x 4 − y 4 = (x 2 )2 − (y 2 )2 = (x 2 − y 2 )(x 2 + y 2 ) Understand the relationship between zeros and factors of a polynomial A-APR 2. Know and apply the Remainder Theorem : For a polynomial 𝑝(𝑥) and a number 𝑎, the remainder on division by 𝑥 – 𝑎 is 𝑝(𝑎), so 𝑝(𝑎) = 0 if and only if (𝑥 − 𝑎) is a factor of 𝑝(𝑥). A-APR 3. Identify zeroes of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-APR 6. Rewrite simple rational expressions in different forms. FUNCTIONS: Analyze functions using different representations F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases, using technology for more complicated cases. F-IF 7c. Graph polynomial functions, identify zeroes when suitable factorizations are available, and showing end behavior. F-IF 8. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret quadratics graphically F-IF 7a. Graph rational functions, identifying zeroes and asymptotes when suitable factorizations are available, and showing end behavior. Pre-Calculus Unit 3 – Exponential and Logarithmic Functions “1” – Emerging I have demonstrated the most basic knowledge/skills relevant to this standard. Scoring Criteria “2” – Developing I have demonstrated relevant knowledge/skills, but have not yet demonstrated convincing evidence of fully meeting the standard. Level 1, plus… Performance Indicators U3PI 8 Graphing exponential and Logarithmic functions U3PI 9 Evaluating common and natural logarithms, properties of logarithms, and solving equations U3PI 10 Exponential and logarithmic application Graph exponential function including the asymptote and any intercepts. Identify the domain of a logarithmic function. Graph logarithmic function including the asymptote and any intercepts. Convert between exponential and logarithmic expressions. Evaluate single-step exponential and logarithmic equations. Setting up and solving basic exponential application problems. CCSS Domains and Standards: NUMBER AND QUANTITY: The Complex Number System N-CN Use complex numbers in polynomial identities and equations. N-CN ALGEBRA: Interpret Structure of expressions A-SSE FUNCTIONS: Analyze functions using different representations F-IF Setting up and solving complex exponential application problems involving continuous compounding. “3” – Achieving I have demonstrated that I have the knowledge/skills defined in the standard. “4” – Excelling I have demonstrated the knowledge/skills defined by the standard with a high level of understanding/ability as defined by the discipline. Level 2, plus… Level 3, plus… Demonstrate Graph natural logarithmic graphically and functions including the algebraically the asymptote and any intercepts. inverse relationship between exponential and logarithmic functions including a comparison of domain and range. Apply properties of logarithms Evaluate multi-step to expand and condense exponential and logarithmic expressions. logarithmic equations. Evaluate single-step natural logarithmic equations. Solve complex logarithmic application problems. Write the exponential or logarithmic equation for a given graph. Solving exponential or logarithmic models requiring calculation of the rate of change.