Honors Pre-calculus Midterm Review: Skills and Sample Problems
§ 1.1 Linear equations, intersection of lines, solving systems of linear equations, distance, midpoint Solutions and graphs of linear equations, x & y-intercepts, equations of horizontal and vertical lines Find the intersection of two or more lines algebraically by solving a system of linear equations
(substitution or elimination) Solutions of a system of linear equations: algebraically, geometrically (graphically)
Find the midpoint and distance of a line segment pp. 5 – 6 # 3, 5, 11, 13, 17, 21, 25, 28, 29
§ 1.2 Slope, parallel, perpendicular: Slopes and equations of parallel and perpendicular lines pp. 11 – 12 # 5 – 7, 9, 13, 17, 19, 21, 23
§ 1.3 Linear equations, perpendicular bisectors
Find equation of a line using point-slope, slope-intercept, or standard forms of a linear equation Find equations of a perpendicular bisector, an altitude, a median
pp. 16 – 18 # 3, 5, 7, 12 – 13, 15, 19a,b, 27 – 28 § 1.4 Linear functions and models
Constructing a linear model from 2 or more data points, or a data point and a rate of change (slope) Finding the solutions (zeros) and breakeven points of linear models
pp. 22 – 25 # 7, 9, 11b, 13, 16, 23
(also breakeven (intersection) points and best buys – fitness and video worksheet problems) § 1.5 Complex numbers (a + bi)
Simplifying, adding / subtracting, multiplying, dividing complex numbers; complex conjugates pp. 28 – 29 # 1 – 27 odd
§ 1.6 Solving quadratic equations: factoring, completing the square, quadratic formula, discriminant losing a root by dividing by a common factor; gaining a root by raising to an even power pp. 35 – 36 # 1, 3, 9, 13, 19, 21, 27, 29, 32, 35a – d, 37, 39, 41
§ 1.7 Quadratic function equations, graphs, intersections of quadratic and linear functions pp. 41 – 42 # 16 – 17, 19 – 20, 23, 27, 29, 31, 33
§ 1.8 Quadratic models
Constructing linear and quadratic models from 3 or more data points using the regression program in the graphing calculator; making interpolation and extrapolation predictions
§ 2.1 Polynomial functions, names by degree and # of terms, leading term, coefficient, finding zeros, roots, evaluation by substitution and by synthetic substitution pp. 56 – 57 # 5, 7, 23, 27
§ 2.2 Polynomial long and synthetic division, remainder and factor theorem, p. 61 # 17, 19, 23
§ 2.3 Graphing polynomial functions, maximums & minimums (extrema), multiple roots long term or end behavior
pp. 66 – 68 # 13, 15, 17, 31 – 32, 37
§ 2.4 Writing polynomial (quadratic and cubic) functions, finding maximums & minimums (extrema)
pp. 71 – 72 # 1, 3, 9 – 11; Cubic # 1, 2
§ 2.6 Solving polynomial equations by factoring (including grouping, quadratic form substitution, sum and difference of cubes), rational root theorem
pp. 83 – 84 # 5, 9, 21, 31
§ 2.7 Polynomial equations, # of roots, complex conjugates, # of irrational roots, # of real roots, sum and product of roots
p. 89 # 1 – 3, 5 – 8
§ 3.1 Linear and absolute value inequalities, solving and graphing inequalities pp. 98 – 99 # 5, 17 – 18, 21, 27, 31 – 32
§ 3.2 Polynomial inequalities (1 variable), sign analysis pp. 103 – 104 # 9, 13, 16, 25
§ 3.3 Polynomial inequalities (2 variables), solving and graphing, including systems of inequalities pp. 106 – 107 # 15, 22, 27, 31
§
Rational functions
Adding, subtracting, multiplying, dividing, and simplifying rational expressions Decomposing rational expression to sum of partial fractions
Solving rational equations
§ 4.1 Relation v. function, solving for y, vertical line test, domain, range, greatest / least integer fn. p. 123 # 9, 11 – 12
§ 4.2 Function operations, composition of functions p. 129 # 19, 23, 27, 32
§ 4.3 Graphs of functions, reflections, symmetry (x-axis, y-axis, y = x, pt. origin), even v. odd fns, cubic functions (point of symmetry, local (or relative) maximum / minimum (or extrema)) pp. 136 – 137 # 3, 5, 12, 15, 21, 29, 31, 33
§ 4.4 Period, amplitude, stretching (shrinking), translating functions pp. 143 – 145 # 1, 5, 7c-f, 8c-f, 9 – 10, 13, 15
§ 4.5 Inverses, one to one functions, horizontal line test
pp. 149 – 150 # 1, 7 – 8, 16 – 17, 2110, 11 – 25 odd § 4.7 Constructing functions in one variable
pp. 161 – 163 # 1 – 2, 7 – 8, 13, 21
§ 5.1 Exponents, laws of exponents, simplifying; exponential growth and decay pp. 173 – 175 # 9, 21, 25, 29, 31, 35 – 37, 41, 47, 49
§ 5.2 Rational (fractional) exponents, more exponential growth and decay
pp. 178 – 180 # 6, 10, 14 – 15, 19, 23, 25, 31, 34, 35a,b, 36a,b, 37, 41, 43, 45, 57 § 5.3 Exponential growth and decay functions; Rule of 72 (doubling); half-life; exponential graphs;
Newton’s Law of Heating and Cooling (notes and problems on handouts, also on web page) pp. 183 – 184 # 7, 11, 13 – 14, 17
§ 5.4 e and the natural exponential function ex; compound interest p. 189 # 5, 7 – 8, 11 – 12, 14
§ 5.5 Logarithms, common log, natural log (ln x), evaluating logs, log graphs, solving equations with logs; also different bases
pp. 194 – 196 # 11 – 17, 25, 27, 29a, 38 – 40
§ 5.6 Logarithm properties (product, quotient, power, reciprocal); consolidating and expanding logs, more solving equations with logs
pp. 200 – 202 # 5, 8, 21 – 25, 27, 33, 45 § 5.7 Solving exponential equations; change of base formula
§ 6.2 Circles, equations, graphing, intersections of a line and a circle pp. 222 – 223 # 7 – 13, 24 – 26a, 33, 39
§ 6.3 Ellipses (center, vertices, foci, major and minor axes), equations, graphing, intersections pp. 228 – 230 # 3, 13 – 14, 18, 25, 31, 35, 40
§ 6.4 Hyperbolas (center, vertices, foci, asymptotes), equations, graphing, intersections pp. 235 – 237 # 1, 5, 7 – 8, 11, 17, 19, 39, 41, 43 – 44
§ 6.5 Parabolas (vertex, focus, directrix), equations, graphing, intersections pp. 240 – 241 # 3, 5, 15, 17, 25, 27, 31
§ 6.6 Systems of quadratic (second-degree) equations, intersections of quadratic relations pp. 244 – 196 # 3, 5, 9, 13, 15, 17
§ 6.7 Eccentricities
c
a
of quadratic relations§ 7.1 Angles: measurement (magnitude & direction), radians, degrees, standard position, coterminal Ðs, reference Ðs, revolutions
pp. 261 – 262 # 1, 3, 5 – 8, 11a, 13a, 17, 27, 32 § 7.2 Sectors of circles: arc length, sector area, apparent size
pp. 264 – 266 # 1 – 13 odd, 14, 18 § 7.3 Sine and Cosine Functions
pp. 272 – 274 # 1 – 7, 16 – 28, 37 – 41
§ 7.4 Sine and Cosine of Special Ðs (30o or 6 , 45o or 4 , 60o or 3 , 90o or 2), Graphs pp. 279 – 281 # 11 – 18, 25, 28, 30
§ 7.5 Tangent and the Reciprocal Trig Functions (cosecant, secant, cotangent) pp. 285 – 286 # 7 – 8, 12 – 18, 23 – 28
§ 7.6 Inverse Trig Functions (Sin–1, Cos–1, Tan–1): graphs; finding Ð measures pp. 289 – 290 # 5 – 9, 11 – 14, 15, 19 – 20
§ 8.1 Trig Equations: Solve simple trig equations
Inclination of a line as it relates to slope and the tangent fn. pp. 299 – 300 # 9, 11, 13, 23, 29, 31, 35
§ 8.2 Sine and Cosine curves: Amplitude, period, horizontal & vertical translations Graph a sine or cosine curve from data,
Write an equation (using sine/cosine) from the graph of a periodic fn. Solving double or triple angle equations
pp. 305 – 306 # 7, 13, 15, 17, 19c, 20b § 8.3 Modeling Periodic Behavior Trig models
pp. 313 – 316 # 3, 13, 15, 25
Graphing calculator skills:
Graph a function in an appropriate window, including solving for y in order to graph quadratic relations Shade an inequality
Enter data into lists (STAT: EDIT)
calculate regression equations (STAT: CALC), including the r and r2 correlation coefficients enter regression equations into y = (VARS: STAT: REG EQ)
plot the points and regression equations (PLOT On, ZOOM 9) make predictions (2nd TRACE: (CALC): VALUE)
Find values, zeros, maxima and minima, intersections (2nd TRACE: (CALC): 1, 2, 3, 4, 5)
Enter numerical values from system of equations into matrices (MATRIX: EDIT)
