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Objectives for Math 115--Precalculus
Students entering Math 115 are expected to be able to
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Find the slope of a line given a line equation or two points
- Write an equation for a line given slope and point or two points
- Solve linear and quadratic equations (including the use of the quadratic formula and factoring)
- Solve radical equations and quadratic types of equations
- Use operations involving polynomial and rational expressions
- Factor polynomials (second-degree or reducible to second-degree, common factors)
- Use the rules for radicals and integer and rational exponents
- Interpret absolute values
- Solve equations involving factored polynomial or rational expressions set equal to 0
- Operate a graphing calculator (enter a function, choose an appropriate window, trace a graph, zoom in, find x-intercepts, evaluate a function or mathematical expression).
Students completing the course should be able to
- Find the domain of functions involving radicals, quotients, logarithms, trig functions and some compositions
- Recognize transformations of basic graphs (absolute value, power, exponential, logarithmic, trigonometric functions) and sketch graphs using vertical and horizontal shifts, reflections in either axis, and vertical stretching or shrinking (horizontal stretching and shrinking only for trig graphs)
- Find x- and y-intercepts for a function
- Find combinations of functions, including compositions
- Recognize from graph when a function inverse exists
- Find the inverse of a function (if it exists)
- Sketch the graph of a quadratic function including the coordinates of the vertex and the intercepts
- Find the maximum or minimum point for a quadratic function
- Solve (factored or factorable) polynomial and rational inequalities using a table of signs
- Solve absolute value inequalities of the form |a + bx| < c, |a + bx| > c, c < | a + bx| < d, where c is any nonnegative number and d is a positive number.
- Recognize polynomials from function or graph; name degree and leading coefficient from function, minimum degree from graph
- Draw the graph of a factored polynomial with correct tail behavior and shape near x-intercepts (without calculator)
- Find a complete graph of a polynomial on a graphing calculator
- Write a polynomial satisfying a description that includes degree, x-intercepts, positive or negative portions, tail behavior and/or y-intercept.
- Find domain, intercepts, vertical and horizontal asymptotes for a rational function
- Draw the graph of a rational function, showing intercepts and asymptotes (if any) (without calculator)
- Rewrite expressions so that they do not contain radicals or negative exponents
- Apply the laws of exponents
- Recognize exponential and logarithmic graphs
- Write an appropriate function to match a given exponential or logarithmic graph
- Graph exponential functions of the form y = d(abx+c), where a > 0
- Graph logarithmic functions of the form y = a logb (x - h) + k where b > 1
- Convert ab = c into the corresponding logarithmic equation loga c = b or vice versa
- Apply the properties and laws of logarithms
- Apply the change-of-base formula to compute loga b when a is not equal to e, 10
- Use the natural number e and the natural logarithm function ln.
- Solve exponential and logarithmic equations
- Solve exponential growth and decay problems
In trigonometry a student should be able to
- Find the terminal point for certain real numbers (e.g., */6, 2*/3)
- Find the terminal point of -t, t + k p** (k any integer) when the terminal point for t is given
[**If you see t + kp, read "pi" for p in this line and the ones that follow.]
- Locate in the correct half of a quadrant the terminal point of a given real number
- Find the reference number for a real number t and use the trigonometric function values of the reference number to find the values for t
- Use the definitions of the six trigonometric functions (of a real number) to find the values for a given t and its terminal point
- Find the exact value of the trigonometric functions for special numbers (multiples of p/6, p/4, p/3, p/2, and p) and numbers coterminal with these special numbers
- Find the exact value of the trigonometric functions when the value and quadrant location of one are given
- Draw the graphs of the six trigonometric functions with different periods or horizontal shifts.
- Draw sine and cosine graphs with different amplitudes, periods, phase shifts, and vertical shifts.
- Find the amplitude, period, and phase shift of the function f(x) = a sin (bx + c) or f(x) = a cos (bx + c).
- Write a function of the form f(x) = a sin (bx + c) or f(x) = a cos (bx + c) to match a given graph.
- Convert angles from degrees to radians or radians to degrees
- Construct an angle in standard position, showing direction and number of turns
- Determine the quadrant in which the terminal side of an angle lies
- Find coterminal angles, including those in the intervals [0, 2p) and (-2p, 0]
- Find arc length, radius or angle when given the other two values
- Use the unit circle, point-in-the-plane, and right triangle definitions of the trigonometric functions
- Use the quotient, Pythagorean, period and odd/even identities to find trig function values and to simplify trigonometric expressions
- Use the sine and cosine addition, subtraction and double-angle identities
- Use the power-reducing identities
- Solve trigonometric equations (exact solutions when special angles are involved, approximations otherwise)
- Solve right triangles when given either a side and an angle or two sides
- Use the Law of Cosines to solve a triangle when given either three sides or two sides and the angle between them
- Use the Law of Sines to solve a triangle when given either two angles and a side or two sides and the angle opposite one of them
- Use the keys sin-1, cos-1, tan-1 to find angles in a triangle either in radians or degrees
- Find the center and radius of a circle from its equation
- Identify and graph a parabola, ellipse, or hyperbola from the standard equation
- Write the equation for a parabola, ellipse, or hyperbola (in standard position) when given a description or graph
- Find the axis of symmetry and vertex of a parabola
- Find the center, major and minor axes and vertices of an ellipse (standard position)
- Find the center, axis where the foci are located and vertices of a hyperbola (standard position)
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