# Precalculus

## Scope and Sequence

Students will be able to:

• Review 6 trigonometric functions of right triangle trigonometry in terms of ratios of sides.
• Evaluate other trigonometric functions given a trigonometric ratio.
• Basic identities – Reciprocal, Pythagorean, cofunction
• Identify angles in standard position
• Identify reference angles for obtuse/non-acute angles
• Use angles as a measure of rotation
• Identify coterminal angles
• Find the arc length subtended by a central angle
• Find the area of a sector subtended by a central angle
• Demonstrate the development of radians as a measure of arc subtended by a central angle
• Find arc length given radians
• Find sector area given radians
• Evaluate trigonometric functions of 30°, 45°, and 60°.
• Evaluate the trigonometric functions of 30°, 45°, and 60° from triangles in the first quadrant of the unit circle.
• Find the measure of reference angles.
• Use the trigonometric values identified in Quadrant 1 to identify the trig values for 120°,135°,150°,210°,225°,240°,300°,315°, and 330°.
• Use the unit circle to identify trigonometric values for
• Convert all unit circle angles to radian measure.
• Connect radian measure to the number line.
• Find coterminal angles using radians.
• Find reference angles using radians.
• Graph the cosine and sine measures for unit circle angles along a number line.
• Define a periodic function.
• Determine the period of sine and cosine.
• Evaluate trigonometric functions of unit circle values without a calculator.
•

Suggested Activities

Students will be able to:

• Identify the roots for sine and cosine.
• Analyze the graphs of sine and cosine to determine if the functions are even or odd or neither.
• Analyze the graphs of sine and cosine to determine maximums and minimums.
• Analyze the graphs of sine and cosine to identify regions where the function is increasing or decreasing.
• Determine if the graphs of sine and cosine are bounded.
• Graph sine and cosine using function transformations.
• Define amplitude in terms of function transformations.
• Identify function transformations that modify the period of sine and cosine.
• Identify and apply horizontal and vertical reflections of sine and cosine.
• Apply horizontal and vertical translations of sine and cosine.
• Generate the function given a transformed sine or cosine graph.
• Using the unit circle, generate all the tangents for the given angles.
• Compare and contrast the period for tangent.
• Graph the tangent values on a number line.
• Express the period for tangent functions.
• Identify any discontinuities for the tangent graph and express as repeating in terms of n.
• Identify the location of any roots for tangent.
• Graph tangent using function transformations.
• Graph reciprocal trigonometric functions along a number line.
• Determine the period for the reciprocal trigonometric functions.
• Identify any discontinuities and express as repeating in terms of n.
• Analyze the graphs of the reciprocal trigonometric functions to identify similarities with the graphs of sin and cosine.
• Use the definition of inverse functions to generate the graphs for arcsin and arccos on a number line.
• Determine the restrictions required to make trigonometric functions and their inverse functions one to one.
• Use an inverse trig function graph to determine an angle measure from a given ratio.

Suggested Activities

Students will be able to:

• Solve simple harmonic motion problems.
• Identify a dampened trigonometric function graph.
• Define angular velocity as rotational speed.
• Convert rotations to degrees and radians.
• Compare angular and linear velocity.
• Express linear velocity in terms of angular velocity.
• Derive Law of Sines
• Apply Law of Sines
• Determine how to identify the ambiguous case.
• Derive Law of Cosines.
• Apply the Law of Cosines.
• Use arc length formulas to derive conversion factor for nautical miles compared to statute miles.
• Determine direction of a vessel using directional bearings by sea.
• Use Heron’s formula to determine area.
• Determine direction of a craft by air using air navigation bearings.

Suggested Activities

Students will be able to:

• Review basic trigonometric identities, reciprocal trigonometric functions, cofunctions, even/odd functions and pythagoran functions.
• Prove an algebraic identity by finding a common denominator.
• Prove trigonometric identities using a graphing calculator.
• Factor functions composed of trigonometric functions.
• Verify trigonometric identities algebraically by manipulating a single side of the identity to transform it into the other.
• Finding the general solution to a trigonometric equation to identify all periodic solutions.
• Use inverse trigonometric functions to identify all possible solutions to a trigonometric equation.
• Graphically verify the formulas for the trigonometric values of sums and differences of angles.
• Evaluate trigonometric expressions using the sum and difference formulas.
• Derive/prove double angle formulas using sum and difference formulas and trigonometric identities.
• Evaluate trigonometric expressions using double angle formulas.
• Derive/prove power reducing formulas using sum and difference formulas and trigonometric identities.
• Evaluate trigonometric expressions using power reduction formulas.
• Evaluate trigonometric expressions using half-angle formulas.
• Evaluate trigonometric expressions using product to sum formulas.
• Solve a trigonometric equation using multiple angle formulas.

Suggested Activities

Students will be able to:

• Define vectors as directed line segments.
• Determine when two vectors are equivalent.
• Find the sum of vectors graphically by tail to head representation and by parallelogram method.
• Find the scalar product of a vector.
• Find the magnitude of a vector.
• Find the direction angle of a vector.
• Write vectors in component form.
• Determine the direction of a given vector.
• Find the sum of vectors in component form algebraically.
• Define Unit Vectors.
• Find Unit vectors in the direction of a given vector.
• Express a given vector as the sum of scalar products of unit vectors.
• Find the dot product of two vectors.
• Use the dot product to identify the angle between two vectors.
• Define orthogonal vectors.
• Determine when two vectors are orthogonal.

Suggested Activities

Students will be able to:

• Define parametric equations and identify situations where they might be encountered.
• Graph parametric equations using technology.
• Generate a table of values to graph parametric equations on the coordinate plane.
• Identify the direction of movement of a parametric variable along a graph on the coordinate plane.
• Convert parametric equations to rectangular equations by eliminating the parametric variable.
• Convert rectangular equations to parametric by defining a parameter.
• Describe projectile motion using parametric equations.
• Simulate horizontal motion using parametric equations.
• Simulate vertical motion using parametric equations.

Suggested Activities

Students will be able to:

• Define polar coordinates as angular movement followed by displacement.
• Plot polar coordinates on a cartesian plan.
• Plot polar coordinates on polar graph.
• Determine all possible polar coordinates for a given point in rectangular coordinates.
• Convert between rectangular and polar coordinates.
• Graph polar equations on a coordinate plane.
• Convert rectangular equations to polar form.
• Convert polar equations to rectangular form.
• Identify common polar graphs including circle, rose curves, lemniscate, cardioid, and limacon.
• Identify equations for common polar graphs.
• Graph polar forms using transformations.

Suggested Activities

Students will be able to:

• Name and identify circles, ellipses, parabolas and hyperbolas from the general form equation.
• Describe the origin of each conic section as sliced from a double napped cone.
• Define each conic section as a locus of points.
• Identify the focus and directrix of both horizontal and vertical parabolas from a graph.
• Identify the focus and directrix of both horizontal and vertical parabolas from an equation.
• Write the equation for a parabola given a graph.
• Graph a horizontal parabola given an equation.
• Explain the definition of an ellipse in comparison to the definition of a circle.
• Determine whether a general equation represents an ellipse.
• Determine whether a conic section is an ellipse or a circle.
• Find the focal axis, foci, vertices, major and minor axes, and semimajor and semiminor axes of an ellipse from a graph.
• Find the focal axis, foci, vertices, major and minor axes, and semimajor and semiminor axes of an ellipse from an equation in standard form.
• Determine the equation of an ellipse given its foci and major/minor axis and center.
• Define the eccentricity of an ellipse and describe its effect on the figures shape.
• Find the eccentricity of an ellipse given a graph.
• Find the eccentricity of an ellipse given an equation in standard form.
• Convert the general equation of an ellipse to standard form.
• Express the definition of a hyperbola as locus of points.
• Explain the definition of a hyperbola, making a connection between the locus definition and the conic equation.
• Determine whether a function is a hyperbola given a graph.
• Determine whether a function is a hyperbola given an equation.
• Find the focal axis, foci, vertices, semitransverse axis, semiconjugate axis, and asymptotes of a hyperbola given a graph.
• Find the focal axis, foci, vertices, semitransverse axis, semiconjugate axis, and asymptotes of a hyperbola given an equation in standard form.
• Find the eccentricity of a hyperbola and describe the effect on its shape.
• Write the equation of a hyperbola given its transverse and conjugate axis’.
• Convert the general equation of a hyperbola to standard form.
• Write the equation of an ellipse that has been rotated by a given angle.
• Identify the equation of a conic section that has been rotated.
• Classify conic sections by their discriminants.

Suggested Activities

Students will be able to:

• Parent function graph review
• Identify the graph and form for constant, linear, quadratic, cubic, square root, cube root, exponential, logarithmic, rational, and step functions.
• Determine the domain and range of constant, linear, quadratic, cubic, square root, cube root, exponential, logarithmic, rational, and step functions.
• Determine the range of rational, polynomial, radical, and power functions graphically and algebraically.
• Determine numerically or graphically whether a function has odd or even symmetry.
• Determine algebraically whether a function has odd or even symmetry.
• Identify whether a function is increasing, decreasing, or constant based on a table of numerical data.
• Identify intervals on which a function is increasing or decreasing graphically.
• Identify absolute extrema from the graph of a function.
• Identify local extrema from the graph of a function.
• Define and identify maximums and minimums as they occur between increasing and decreasing regions for all parent functions.
• Define boundedness for a function.
• Determine the boundedness of all parent functions.
• Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
• Identify the end behavior of a function.
• Express end behavior using limit notation.
• Differentiate between infinite limit and one sided limit notation.
• Identify the horizontal and vertical asymptotes of a graph.
• Express horizontal and vertical asymptote using limit notation.
• Determine graphically whether a function is continuous or discontinuous at a point.
• Classify the type of discontinuity graphically.
• Identify and graph piecewise functions.
• Analyze piecewise functions.
• Evaluate piecewise functions given an input.
• Determine graphically and algebraically whether a piecewise function is continuous or discontinuous at a point.
• Define and determine the composition of two given functions
• Find the domain of the composition of two given functions.
• Decompose a given function as the composition of simpler functions.
• Analyze a real-world situation of composition of functions.
• Identify and verify implicitly two functions given a relation.
• Determine if a function is one-to-one using the Horizontal Line Test.
• Determine an inverse function graphically and algebraically. Graph an inverse by exchanging the coordinates of the points.
• Verify that two functions are inverses of one another algebraically and graphically.
• Compare domain and ranges of the function and its inverse.
• Determine if the inverse is a relation or a function.
• Restrict the domain of a function such that it will be invertible.
• Recognize function transformations written in af(b(x-c))+d form.
• Apply function transformations written in af(b(x-c))+d form to a given parent function.
• Write the equation of a function under a horizontal or vertical reflection when described graphically or verbally.
• Write the equation of a function under a vertical or horizontal stretch or shrink when described graphically or verbally.
• Graph the equation of a function under multiple transformations.
• Write the equation of a function under multiple transformations.

Suggested Activities

Students will be able to:

• Compare graphs of power functions with differing powers.
• Identify the power and constant of variation of the power function.
• Determine graphically and algebraically which power functions are even and which are odd.
• Relate the end behavior of a polynomial to the sign of the leading coefficient and its power - leading term test
• Translate between verbal and algebraic descriptions of power functions.
• Analyze key features of a power function: domain, range, continuity, symmetry, asymptotes, min/max, and end behavior.
• Express end behavior using limit notation.
• Sketch even powered polynomial functions with key components.
• Identify potential number of roots for a quadratic and expand to higher powered even polynomials.
• Apply the leading term test to determine the orientation of the graph and end behavior.
• Express the end behavior of the polynomial using limit notation.
• Identify regions of increase and decrease.
• Identify local and absolute maximum and minimum values graphically.
• Find the vertex and roots of a quadratic equation algebraically.
• Solve a polynomial with imaginary roots using the quadratic formula.
• Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
• Identify and approximate irrational zeroes of a quadratic using a graphing calculator.
• Identify the multiplicity of a factor.
• Determine the effect of multiplicity on the behavior of a root.
• cubics review/odd degree
• Determine the minimum number of possible roots for an odd powered polynomial.
• Sketch odd powered polynomial functions with key components.
• Identify the potential number of roots for a cubic function and expand to higher powered odd polynomials.
• Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
• Identify and approximate irrational zeroes of a quadratic using a graphing calculator.
• Apply the leading term test to determine the orientation of the graph and end behavior.
• Express the end behavior of the polynomial using limit notations.
• Identify regions of increase and decrease.
• Identify local and absolute maximum and minimum values graphically.
• Find the roots of a cubic function by factoring by grouping.
• Identify the multiplicity of a factor.
• Determine the effect of multiplicity on the behavior of a root.
• Generate a polynomial by converting roots to factors and multiplying.
• Identify the multiplicity of a factor before expanding.
• Determine a polynomial given its zeroes.
• Expand a power of a binomial expression using the Binomial Theorem.
• Generate the coefficients for a binomial expansion using Pascals triangle.
• Generate the coefficients for a binomial expansion using the binomial coefficient formula.
• Convert imaginary and irrational roots to factor form and multiply to generate a polynomial function with real coefficients.
• Find the roots of a perfect cube binomials by using the formula for sums and differences of perfect cubes.
• Identify the exact number of complex zeroes of a given polynomial.
• Determine whether the complex conjugate of a zero is also a zero of a polynomial.
• Find the quotient of two polynomials using long division.
• Given a root, use long division to identify other roots for a polynomial.
• Find the quotient of two polynomials using synthetic division.
• Given a root, use synthetic division to identify other roots for a polynomial.
• Use the Remainder Theorem to evaluate functions at a given value.
• Identify all possible rational roots for a polynomial using the Rational Root Theorem (p/q).
• Determine possible number of positive and negative roots using Descartes Rule of Signs.
• Determine which possible rational roots are actual roots using synthetic division.
• Express a polynomial as a product of linear and irreducible quadratic factors.
• Solve a polynomial completely using long division, synthetic division, factoring and the quadratic formula to identify all roots.

Suggested Activities

Students will be able to:

• Sketch the graph of a rational function as transformations of 1/x.
• Identify vertical and horizontal asymptotes.
• Explain the difference between a vertical and horizontal asymptote.
• Identify any holes that occur.
• Describe the end behavior of a rational function using limit notation.
• Describe the left and right side behavior around any discontinuities using limit notation.
• Determine the domain of a rational function.
• Compose sums and differences of rational functions by finding a common denominator.
• Analyze the graph of complex rational functions.
• Identify the roots of a rational function.
• Find any vertical asymptotes for a rational function.
• Determine when a rational function will have no vertical asymptotes.
• Identify rational functions that will have oblique asymptotes.
• Find the equation of oblique asymptotes using long division.
• Identify the domain and range of complex rational functions.
• Describe the end behavior of complex rational functions using limit notation.
• Sketch the graph of rational functions.
• Solve a rational equation by clearing fractions.
• Determine whether solutions are valid or extraneous.
• Use rational functions to solve real world applications.

Suggested Activities

Students will be able to:

• Determine the domain over which solutions to a polynomial are zero, positive and negative graphically.
• Identify critical values for a polynomial.
• Explain the effect even multiplicity has on a factor in a sign chart.
• Determine the domain over which solutions to a polynomial are zero, positive and negative algebraically using a sign chart.
• Sketch a graph for a polynomial inequality using a sign chart.
• Solve a polynomial inequality using a sign chart.
• Determine the domain over which solutions to a rational function are undefined, zero, positive and negative graphically.
• Identify critical values for a rational function.
• Determine the domain over which solutions to a rational function are undefined, zero, positive and negative algebraically using a sign chart.
• Sketch a graph for a rational function inequality using a sign chart.
• Solve a rational inequality using a sign chart.
• Determine the domain over which solutions to a square root function are undefined, zero, positive and negative graphically.
• Identify critical values for a square root function.
• Determine the domain over which solutions to a square root function are undefined, zero, positive and negative algebraically using a sign chart.
• Sketch a graph for a square root function inequality using a sign chart.
• Solve a square root inequality using a sign chart.

Suggested Activities

Students will be able to:

• Review Exponential Graphs and Equations
• Identify and graph an exponential function using technology.
• Identify and sketch the graph of an exponential function using a table of values.
• Sketch the graph of an exponential function using transformations.
• Identify key attributes of an exponential graph from an equation, including y-intercept, horizontal asymptote and end behavior.
• Express end behavior of an exponential function using limit notation.
• Evaluate an exponential function.
• Differentiate between graphs and equations of exponential growth and decay.
• Review Logarithmic Graphs and Equations
• Identify and graph a logarithmic function using technology.
• Explain the inverse relationship between logarithmic and exponential functions.
• Identify key attributes of logarithmic graphs including roots, y-intercepts and asymptotes.
• Express end behavior of logarithmic graphs using limit notation.
• Evaluate a logarithmic expression.
• Convert a logarithmic expression to an exponential expression.
• Convert an exponential expression to a logarithmic expression.
• Simplify and expand logarithmic expressions using properties of logarithms.
• Define the natural logarithm.
• Define the common logarithm.
• Evaluate a logarithm by changing the base.
• Simplify a natural logarithm using properties of logarithms and laws of exponents.
• Find rates of growth and decay for exponential functions given real world applications.
• Find the half life of a radioactive substance.
• Model population growth using exponential functions.
• Model a cooling problem using Newton’s Law of Cooling.
• Identify and evaluate a logistic function.
• Solve an exponential equation using a logarithm.
• Solve a logarithmic equation.
• Model economic growth and decay given real world applications.
• Compute interest compounded annually, quarterly, and monthly.
• Compute interest compounded continuously.
• Compute annual percentage yield (APY).
• Compute present value for an amount with a given interest rate.

Suggested Activities

Students will be able to:

• Define a sequence explicitly.
• Graph a sequence using technology.
• Explain the difference between a sequence and a series.
• Express the end behavior of a sequence using limit notation.
• Generate a non-arithmetic, non-geometric sequence.
• Determine whether a given sequence is arithmetic.
• Define an arithmetic sequence explicitly.
• Generate an arithmetic sequence.
• Determine whether a sequence is geometric.
• Define a geometric sequence explicitly.
• Identify compound interest as a type of geometric sequence.
• Generate a geometric sequence.
• Define a partial sum of an infinite sequence.
• Write out the sequence of partial sums for a series.
• Find the nth partial sum of an arithmetic sequence.
• Find the nth partial sum of a geometric sequence.
• Express the nth partial sum of an arithmetic sequence using sigma notation.
• Express the nth partial sum of a geometric sequence using sigma notation.
• Evaluate a sum expressed in sigma notation.
• Determine the limit of an infinite sequence from the sequence of partial sums.
• Determine if an infinite sequence will converge or diverge.
• Determine the conditions for a geometric sequence to converge.
• Find the sum of an infinite geometric series if it exists.
• Demonstrate mathematical induction by showing that substituting k+1 into a formula generates the same answer as substituting k and then adding on a k+1 term.

Suggested Activities