Mastery. PRECALCULUS Student Learning Targets

Transcription

1 PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions, Formulas, Sample Problems, etc. Struggles What I need help on) Write the a n formula for a sequence. Mastery 2. I can represent sequences and series using various notations. a n formula, sigma notation - (means to add up the terms) 3. I can identify arithmetic and geometric sequences and series. 4. I can discover and justify the formula for a finite arithmetic series. 5. I can discover and justify the formulas for finite and infinite geometric series. Big Idea: Graphing and Analysis of Functions 6. I can graph polynomial, rational, piece-wise, exponential, and logarithmic functions. 7. I can identify the effects of changing the parameter a in y = af(x), y = f(ax), y = f(x a), and y = f(x) + a, given the graph of y = f(x). Arithmetic: a n = dn + c where d is the common difference, n is the term number, and c is a constant (arithmetic: add the same number to get to the next term) Geometric: a n = a 1 r (n-1) where r is the common ratio and a 1 is the first term (multiply the same number to get to the next term) Sum of an arithmetic sequence (an arithmetic series): S n = (n/2)(a 1 + a n ) - (number of terms divided by 2 and then multiplied by first term plus last term) Sum of infinite geometric sequence (an infinite geometric series): S = a 1 /(1-r) where -1< r <1 - (first term divided by 1mius the common ratio) Sum of infinite geometric sequence: S n = a 1 (1- r n )/(1-r) Polynomial: Find x and y intercepts ( plug in zero for x or y and solve for the other variable). Determine multiplicity of each x-intercept. Evaluate end behavior - plot extra points if needed and sketch the graph. Rational: Determine domain - Find vertical and horizontal asymptotes - plots points and sketch the graph. Piece-wise: Graph each piece just on the x values for that section. Exponential: Determine horizontal asymptote (unless it s shifted vertically, this is at y = 0), plot the main points (plug in 0 & 1 for x), sketch the graph. Logarithmic: Determine the vertical asymptote (unless it s shifted horizontally, this is as x = 0), plot the main points (plug in 0 & 1 for y), sketch the graph. Transformations: In y = a(x - h) 2 + k, what does the a, h, & k do to the graph of y = x 2? (Vertical shifts: k, horizontal shifts: h, stretches & shrinks: a, and flips: negatives)

2 8. I can identify the domain, range, and other attributes of families of functions and their inverses. 9. I can approximate instantaneous rates of change & find average rates of change using graphs or numeric data. 10. I can identify and analyze continuity, end behavior, asymptotes, symmetry (odd and even functions), and limits, and connect these concepts to graphs of functions. 11. I can determine intervals over which a function is increasing or decreasing, and describe the intervals using interval notation. Domain - x values on the graph (sort-of the width of the graph) Range - y values on the graph (sort-of the height of the graph) Inverse - switch the x s and y s and solve for the new y. Average rate of change is just SLOPE. m = (y 2 - y 1 )/(x 2 - x 1 ) Instantaneous rate of change is when two points are really close together on a graph and you do the slope, it s getting closer to what a slope of a tangent line would be. Continuity: Is their any holes or jumps in the graph or is it contiuous? End behavior: As x or as x -, what does y go to? Polynomials: even power : both ends go same direction (both ends go up if coefficient is positive & both down if coefficient is negative), odd power :opposite directions. Rational Functions: Vertical & Horizontal Asymptotes (V.A: set denominator equal to zero. H.A: compare powers - top higher = no H.A., bottom higher = H.A. is y = 0 Equal powers = make fraction out of coefficients) Decreasing - going down as you go to the right Increasing - going up as you go to the right Describe increasing and decreasing intervals using the x values and interval notation Big Idea: Solving Polynomial Functions 12. I can raise a binomial to a power using the Binomial Theorem and Pascal s Triangle. 13. I can determine the number and nature of solutions to polynomial equations with real coefficients over the complex numbers. Pascal s Triangle 1 These are the coefficients of a 1 1 binomial expansion - then the variable powers go in descending etc.. or ascending order. (You can find a specific term by using nc r instead of Pascal s triangle.) Solutions, zeros, x-intercepts, & roots are the same thing if real numbers. Solutions can be real or imaginary (with the i s). Multiplicity is how many times a factor appears: (x - 1) (x + 3) (x + 3) = 0 has two solutions: x = 1 with multiplicity of 1 and x = -3 with multiplicity of 2.

3 14. I can factor polynomials to solve equations. Factoring: 1. GCF 2. Number of terms: 2 terms look for difference of squares that factor to congugates, 3 terms quadratic factoring, 4 terms look for Grouping 3. Do p/q s list and do synthetic division until you find one that gives you zero for the remainder. Keep going until you get down to something you can factor - like a 2 or 3 term one. 15. I can understand the relationships among the solutions of a polynomial equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial. Example: Factors: f(x) = (x - 3) (x + 4) (x - 7) (x - 6i) (x + 6i) Solutions, zeros, roots: x = 3, -4, 7, 6i, -6i x-intercepts: x = 3, -4, I can write an equation with given solutions. Example: Given solutions x = 8, 4, 2i, write a polynomial equation: (x - 8) (x - 4) (x - 2i) (x + 2i) (multiply it out to get the polynomial equation) (Imaginary solutions come in conjugate pairs, so -2i is also a solution) Big Idea: Model Real-world Relationships with Matrices 17. I can represent real-world situations with matrices. Story problems using matrices (Woos Bakery, etc. - adding, subtracting or multiplying matrices) 18. I can add, subtract, and multiply (including scalar multiplication) matrices using paper and pencil, and computer programs or calculators. 19. I can demonstrate that matrix multiplication is associative and distributive, but not commutative. 20. I can determine additive and multiplicative identities and inverses of a matrix when they exist. 21. I can solve systems of linear equations with up to three variables using matrices. On calculator, go to matrix menu and edit to type them in. Do multiplications, etc. on main calculator screen. Multiply rows of the first matrix by columns of the second one, (so the numbers of rows in the first matrix must equal the number of columns in the second) Must be a square matrix to have an inverse. On the calculator: enter in the matrix and then use the x -1 key. On the calculator: Enter the matrix and then under the matrix - math menu - use rref.

4 Big Idea: Trigonometric Definitions, Functions, and Graphing/Solving Trigonometric Equations 22. I can define the six trigonometric functions using the unit circle. 23. I can prove trigonometric identities using definitions, the Pythagorean Theorem, or other relationships. 24. I can construct the graphs of the trigonometric functions and their inverses, and describe their behavior, including periodicity and amplitude. 25. I can simplify trigonometric expressions and solve trigonometric equations using identities. 26. I can solve problems using the Law of Sines and the Law of Cosines. On the unit circle: y-coordinate is sin, x-coordinate is cos, y/x is tan, secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent. Reciprocal, quotient, Pythagorean, double angle identities, etc. Basic 6 trig functions: amplitude, vertical shift, period, and phase shift. (Normal period of trig functions is 2 unless it is tangent or cotangent, then it s.) Solving trig equations: Either isolate the trig part or set it all equal to zero and factor. Look up the solutions (angles) on the unit circle. To solve a right triangle: use Pythagorean Theorem and SOH CAH TOA. To solve other triangles: use Law of Sines and Cosines (Use Law of Sines when you have an angle and its opposite side given) Big Idea: Conic Sections 27. I can write equations of conic sections in standard form. 28. I can identify the geometric properties of conic sections (i.e., center point, vertices, asymptotes). Circle: (x - h) 2 + (y - k) 2 = r 2 : (h, k) is the center and r = radius Ellipse: (x - h) 2 / a 2 + (y - k) 2 / b 2 = 1 Hyperbola: (x - h) 2 / a 2 - (y - k) 2 / b 2 = 1 Find the center (h, k) and then count left/right & up/down to find vertices, etc. Sketch the ellipse or draw the box and asymptotes then sketch the hyperbola (test a point to see which way to draw it - up and down or left and right) Big Idea: Parametric and Polar Equations

5 29. I can define a curve parametrically and draw parametric graphs. 30. I can define and use polar coordinates and relate them to Cartesian coordinates. Either plot points (using 3 columns: t, x, & y, then plot the (x, y) points) Or eliminate the parameter (the t) by solving one equation for t and substituting into the other equation. Then you just have an x & y equation, so just graph it using techniques we already know. Instead of using (x, y) points, use (r, ) points, where r is the radius and angle from the initial side (positive x-axis) is the 31. I can graph polar equations on the calculator. Change mode on the calculator to polar and enter the graphs. To graph a circle, just enter r = a number. Example: circle centered at the origin with radius 3 is just r = 3. Big Idea: Linear Regression 32. I can use the lists on a calculator to enter sets of data and fit it to a regression line. 33. I can compute predictions of y-values for given x-values using a regression equation, and recognize the limitations of such predictions. On the calculator: go to STAT menu and then EDIT: Enter x values in list 1 and y values in list 2. Then go to STAT menu and then CALC and do Linreg. It will give you the a & b values for a linear equation y = ax + b, where a is the slope and b is the y-intercept. Graph this equation and use the trace button to see what the graph is doing at different values. Use the trace button to see the behavior of the graph and analyze or make predictions. Big Idea: Discrete Probability

6 34. I can compute probabilities using areas under the Normal Curve. Average (mean): Standard Deviation: