Subject/Grade Level: Algebra/6th Grade



Grade Level: 11th and 12th

Title: Pre-Calculus – Functions & Their Graphs

Time Frame: 4 weeks

Enduring Understanding: The graph of a function is a visual aid to understanding the function.

Essential Questions: What is a function and what is the meaning of its domain and range? What operations may be performed on functions? How are those operations carried out? What situations can be modeled with functions? How can functions be used to model and solve problem situations?

| | |Vocabulary Development | |Instructional Activities/ | |

|Concept/Topic |Indicators | |Assessment Strategies |Extension Activities |Resources |

| | |Slope | | | |

|Functions |PC-2.1 Carry out a procedure to graph |Parent function | | | |

| |parent functions (including y = |Quadratic | | | |

|Graphs |xn, y = loga x, y = ln x, y = [pic], y |Absolute Value | | | |

| |= ex, y = ax, y = sinx, y = cos x, y |Piecewise functions | | | |

| |= tan x, y =csc x, y = sec x, and y | | | | |

| |= cot x). | | | | |

| |PC-2.2 Carry out a procedure to graph transformations (including |Transformation | | | |

| |–f(x), a • f(x), f(x) + d, |Reflection | | | |

| |f(x - c), f(-x), f(b • x), |f(x)|, and f(|x|)) of parent functions |Combination | | | |

| |and combinations of transformations. | | | | |

| |PC-2.3 Analyze a graph to describe the transformation (including | | | | |

| |–f(x), a • f(x), f(x) + d, | | | | |

| |f(x - c), f(-x), f(b • x), |f(x)|, and f(|x|)) of parent functions. | | | | |

| | | | | | |

| |PC-2.5 Analyze graphs, tables, and equations to determine the domain |Domain | | | |

|Domain and Range |and range of parent functions or transformations of parent functions |Range | | | |

| |(including y = xn, y = loga x, y = ln x, y = [pic], y = ex, y = ax, y|Function | | | |

| |= sin x, y = cos x, y = tan x, y = csc x, y = sec x, and y = cot x). |Vertical Line test | | | |

| |PC-2.6 Analyze a function or the symmetry of its graph to determine |Even | | | |

|Describe symmetry of even and odd|whether the function is even, odd, or neither. |Odd | | | |

|functions | | | | | |

|Inverse functions |PC-2.8 Carry out a procedure to determine |Inverse | | | |

| |whether the inverse of a function |Horizontal line test | | | |

| |exists. |One-to-one | | | |

| |PC-2.9 Carry out a procedure to write a rule | | | | |

| |for the inverse of a function, if it | | | | |

| |exists. | | | | |

Title: Pre-Cal - Polynomials

Time Frame: 4 weeks

Enduring Understanding: Knowledge of exponents will be used to manipulate and solve polynomials.

Essential Questions: How is factorability of a polynomial determined? How are polynomials divided using synthetic and long division? What are zeros of a functions and what methods are there to find them? What are the important characteristics of a polynomial graph and how are those determined? How can those important characteristics help to solve real world problem situations?

|Polynomial functions |PC-3.1 Carry out a procedure to graph quadratic and higher-order |Quadratic function | | | |

| |polynomial functions by analyzing intercepts and end behavior. |Axis of symmetry | | | |

| |PC2.7 Recognize and use connections among significant points of a |Standard form | | | |

| |function (including roots, maximum points, and minimum points), |Vertex form | | | |

| |the graph of a function, and the algebraic representation of a |Relative Max | | | |

| |function. |Relative Min | | | |

| | |Absolute Max | | | |

| | |Absolute Min | | | |

| | |Root/zero | | | |

| | |Increasing | | | |

| | |Decreasing | | | |

| | |Constant | | | |

| | | | | | |

| |PC-3.2 Apply the rational root theorem to determine a set of | | | | |

| |possible rational roots of a polynomial equation. | | | | |

| |PC-3.3 Carry out a procedure to calculate the zeros of polynomial | | | | |

| |functions when given a set of possible zeros. | | | | |

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| |PC-3.5 Analyze given information to write a polynomial function | | | | |

| |that models a given problem situation. | | | | |

| |PC-3.6 Carry out a procedure to solve polynomial equations | | | | |

| |algebraically. | | | | |

| |PC-3.7 Carry out a procedure to solve polynomial equations | | | | |

| |graphically. | | | | |

| | | | | | |

|Polynomial |PC-3.10 Carry out a procedure to solve polynomial inequalities |Test intervals | |Appendix B.4 | |

|inequalities |algebraically. | | | | |

| |PC-3.11 Carry out a procedure to solve polynomial inequalities | | | | |

| |graphically. | | | | |

Title: Pre-Calculus – Rational functions

Time Frame: 2 weeks

Enduring Understanding: Rational functions can be represented verbally, numerically, graphically, and analytically to understand patterns and relationships.

Essential Questions: What are rational functions? What is asymptotic behavior? What situations are modeled using a rational function and how can these be used to solve problems?

|Rational Functions |PC-3.4 Carry out procedures to determine characteristics of |Rational function | | | |

| |rational functions (including domain, range, intercepts, |Asymptote | | | |

| |asymptotes, and discontinuities). | | | | |

| | | | | | |

| |PC-3.8 Carry out a procedure to solve rational equations | | | | |

| |algebraically. | | | | |

| |PC-3.9 Carry out a procedure to solve rational equations | | | | |

| |graphically. | | | | |

Title: Pre-Calculus – Exponential and Logarithmic Functions

Time Frame: 2 weeks

Enduring Understanding: Exponential and logarithmic functions are inverse functions.

Essential Questions: What are the characteristics of exponential and logarithmic functions? How are exponential and logarithmic functions simplified, solved and manipulated? What situations are modeled with exponential and log equations? How can these functions be used to solve problems?

|Exponential Functions: |PC-4.1 Carry out a procedure to graph exponential functions by |Exponential function | | | |

| |analyzing intercepts and end behavior. | | | | |

| |PC-4.3 Carry out procedures to determine characteristics of | | | | |

| |exponential functions (including domain, range, intercepts, and | | | | |

| |asymptotes). | | | | |

|Logarithmic Functions: |PC-4.2 Carry out a procedure to graph logarithmic functions by |Logarithmic function | | | |

| |analyzing intercepts and end behavior. |Natural logarithm | | | |

| |PC-4.4 Carry out procedures to determine characteristics of | | | | |

| |logarithmic functions (including domain, range, intercepts, and | | | | |

| |asymptotes). | | | | |

|Solving exponential and log |PC-4.7 Apply the laws of logarithms to solve problems. | | | | |

|functions |PC-4.5 Apply the laws of exponents to solve problems involving | | | | |

| |rational exponents. | | | | |

| |PC-4.8 Carry out a procedure to solve | | | | |

| |exponential equations algebraically. | | | | |

| |PC-4.9 Carry out a procedure to solve exponential equations | | | | |

| |graphically | | | | |

| |PC-4.10 Carry out a procedure to solve logarithmic equations | | | | |

| |algebraically. | | | | |

| |PC-4.11 Carry out a procedure to solve | | | | |

| |logarithmic equations | | | | |

| |graphically | | | | |

|Problem situations |PC-4.6 Analyze given information to write an exponential function | | | | |

| |that models a given problem situation. | | | | |

| | | | | | |

Title: Pre-Calculus – Trigonometric Functions Time Frame:

Time Frame: 4 weeks

Enduring Understanding: Trigonometry can be used to determine indirect measurements of lengths and angles

to solve a variety of problems.

Essential Questions: What are the six trigonometric functions and what do their graphs look like? How are trigonometric equations used to model situations and solve problems? What are the trigonometric identities and how can they be applied to solve equations?

|Trigonometric Functions |PC-5.1 Understand how angles are measured in either |Angle | |Trig on a Paper Plate | |

| |degrees or radians. |Initial side | | | |

| |PC-5.2 Carry out a procedure to convert between degree and|Terminal side | | |

| |radian measures. |Vertex | | |m |

| |PC-5.6 Apply a procedure to evaluate trigonometric |Standard position | | | |

| |expressions. |Positive angle | | | |

| | |Negative angle | | | |

| | |Central angle | | | |

| |PC-5.8 Analyze given information to write a trigonometric | | | | |

| |equation that models a given problem situation involving | | | | |

| |right triangles. | | | | |

| |PC-5.4 Carry out a procedure to graph trigonometric |Amplitude | | | |

| |functions by analyzing intercepts, periodic behavior, and |Period | | | |

| |graphs of reciprocal functions. |Phase shift | | | |

| |PC-5.5 Carry out procedures to determine the | | | | |

| |characteristics of trigonometric functions (including | | | | |

| |domain, range, intercepts, and asymptotes). | | | | |

|Solving Trig Equations |PC-5.10 Carry out a procedure to solve trigonometric | | | | |

| |equations algebraically. | | | | |

| |PC-5.11 Carry out a procedure to solve trigonometric | | | | |

| |equations graphically. | | | | |

|Problem solving |PC-5.7 Analyze given information | | | | |

| |to write a trigonometric | | | | |

| |function that models a | | | | |

| |given problem situation | | | | |

| |involving periodic | | | | |

| |phenomena. | | | | |

|Inverse Trig Functions |PC-5.13 Apply a procedure to graph |Inverse trig function | | | |

| |the inverse functions of | | | | |

| |sine, cosine, and tangent | | | | |

|Polar Coordinates |PC-5.3 Carry out a procedure to plot points in the polar |Polar coordinate system | | | |

| |coordinate system. |Pole | | | |

| | |Polar axis | | | |

| | |Directed angle | | | |

| | |Directed distance | | | |

|Law of sines and cosines |PC-5.12 Apply the laws of sines and cosines to solve | | | | |

| |problems. | | | | |

|Area of a triangle |PC-5.9 Carry out a procedure to calculate the area of a | | | | |

| |triangle when given the lengths of two sides and the | | | | |

| |measure of the included angle. | | | | |

|Solve problems |PC-5.15 Carry out a procedure to |Angle of inclination | | | |

| |compute the slope of a line | | | | |

| |when given the angle of | | | | |

| |inclination of the line. | | | | |

Title: Conic Sections

Time Frame: 2 weeks

Enduring Understanding: Conic sections can be identified and graphed from their equations.

Essential Questions: What are the conics? How are they formed? What are the differentiating characteristics of each conic? What are the applications of the conics in our world?

|Circles |PC-6.1 Carry out a procedure to graph the circle whose | | | |Algebra II McDougal Littell Section 10.3 |

| |equation is the form [pic]. | | | | |

| |PC-6.2 Analyze given information about the center and | | | | |

| |the radius or the center and the diameter to write an | | | | |

| |equation of a circle. | | | | |

| |PC-6.3 Apply a procedure to calculate the | | | | |

| |coordinates of points where a line | | | | |

| |intersects a circle. | | | | |

|Ellipse |PC-6.4 Carry out a procedure to graph the ellipse whose| | | |Algebra II McDougal Littell Section 10.4 |

| |equation is the form [pic]. | | | | |

|Hyperbola |PC-6.5 Carry out a procedure to graph the hyperbola | | | |Algebra II McDougal Littell Section 10.5 |

| |whose equation is the form [pic]. | | | | |

|Parabola |PC-6.6 Carry out a procedure to graph the parabola | | | |Algebra II McDougal Littell Section 10.2 |

| |whose equation is the form [pic]. | | | | |

| | | |Course Assessment | |Course Resources |

| | | |Houghton Mifflin Test Generator V | | |

| | | |7.0 (provided with textbook) | | |

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