Packet #1: Precalculus Review
BASIC SHAPES: Match each function with its basic shape. Use a graphing calculator only if absolutely necessary.
______1. A.B.C.
______2.
______3.
______4.
______5.D.E.F.
______6.
______7.
______8.G.H.I.
______9.
INTERCEPTS
y-intercept(s): set x= 0 and solve for y.
x-intercept(s): set y= 0 and solve for x.
10. Find any intercepts for each graph.
a. b. c.
HW: p. 8 #s 1-4all, #s 17-23odd
THE TRANSLATION POINT (TP)
(h, k)-form: Suppose, then the point (h, k) is the translation point (TP).
[Note: to correctly establish the TP, the coefficient of x MUST BE positive one]
11. Identify the TP for each graph. [you may need to manipulate the function first]
a. b. c.
d. e. f.
SKETCHING BASIC SHAPES USING CUES
12. Sketch: by first answering the cues.
BS:
TP:
y-int:
x-int:
*confirm with your graphing calculator when done.
13. Sketch: by first answering the cues.
BS:
TP:
y-int:
x-int:
*confirm with your graphing calculator when done. HW: p. 8 #s 5-11odd
BASIC SHAPES WITH TRANSFORMATIONS
Match each progression with an appropriate transformation. Use your graphing calculator if you’re not sure how the original function is being transformed.
14. If , then is a ______.A. horizontal reflection
15. If , then is a ______.B. horizontal stretch
16. If , then is a ______.C. vertical reflection
17. If , then is a ______.D.horizontal compression
18. If , then is a ______.E. vertical compression
19. If , then is a ______.F.vertical stretch
SKETCHING BASIC SHAPES USING CUES (ADVANCED)
20. Sketch: by first answering the cues and describing any compressions or stretches.
BS:
REF:
TP:
y-int:
x-int:
*confirm with your graphing calculator when done.
Something to think about: Can any graph be sketched using TRANSFORMATIONS?
SYMMETRY
Some graphs exhibit symmetry over the x-axis, the y-axis, or both (origin symmetry).
21. Complete each sketch so that it exhibits the type of symmetry named.
a.b.c.
ORIGIN Y-AXIS X-AXIS
ALGEBRAIC SYMMETRY TESTS
Y-AXIS: when replacing all x’s with –x yields the original problem [called EVEN functions]
X-AXIS: when replacing all y’s with –y yields the original problem (*these are not functions)
ORIGIN: when replacing all x’s and y’s with –x and –y yields the original problem [called ODD functions]
22. Test for symmetry with respect to each axis and to the origin.
a.b. c.
d. e. f.
SYMMETRY – ANOTHER PERSPECTIVEHW: p. 8 #s 25-25odd
A function is EVEN if p. 8 #s 41, 45, 49, 53
A function is ODD if
23. Tell whether the graphs from question #22 are EVEN, ODD, or NEITHER.
POINTS OF INTERSECTION
24. Find the point(s) of intersection algebraically, using either elimination or the substitution method.
a. b. c.
d. e.
25. Use your graphing calculator’s intersection feature to determine the vertices of the triangle formed by
the three lines given:
, and
HW: p. 9 #s 63-71odd
SLOPE (possibly for the last time in your life!)
26. Complete:
a. If a line is increasing, then its slope will be ______.
b. If a line is decreasing, then its slope will be ______.
c. If a line is horizontal, then its slope will be ______.
d. If a line is vertical, then its slope will be ______.
27. Find the slope of the line passing through the points named.
a. (1, 2) and (-2, 4)
b.
28. Use the point on the line and the slope of the line to find three additional points that the line passes
through. (There is more than one correct answer.)
a. (-3, 4); m is undefinedb. (-2, -2); m = 2
HW: p. 16 #s 1-5odd, 9-17odd
EQUATIONS OF LINES
Point-slope Form:
Slope-intercept Form:
Standard Form:, where a> 0 and a, b, and c are integers
General Form:, where a> 0 and a, b, and c are integers
Vertical Line:, where c is any real number
Horizontal Line:,where c is any real number
29. Find the slope and y-intercept of each line.
a. b.
HW: p. 16 #s 23, 25
30. Find an equation of the line that passes through each point and has the indicated slope.
a. (-1, 2); m is undefined [general form]
b. (0, 4); m = 0 [standard form]
c. (-2, 4); m = -3/5 [slope-intercept form]
HW: p. 16 #s 27-31odd
31. Find the equation of the line that passes through the points.
a. (-3, -4) and (1, 4) [point-slope form]
b. (-3, 6) and (1, 2) [general form]
c. (1, -2) and (3, -2) [standard form]
HW: p. 16 #s 35, 39, 41
32. Use your graphing calculator to find the points of intersection between the two parabolas. Then find the
equation of the line through the points of intersection in slope-intercept form.
and
PARALLEL AND PERPENDICULAR LINES
If two lines are parallel, then their slopes are ______.
If two lines are perpendicular, then their slopes are ______.
33. Write an equation of the line through the given point (a) parallel to the given line and (b) perpendicular
to the given line.
- (-3, 2);
- (-6, 4);
- (-1, 0);
HW: p. 17 #s 59-63odd
EVALUATING A FUNCTION
34. Suppose , evaluate:
a. b. c.
35. Suppose , evaluate:
a. b. c.
36. Suppose , evaluate:
a. b. c.
HW: p. 27 1-9odd
DOMAIN OF A FUNCTION
The denominator of any rational expression cannot be 0
If you have a radical of even index, then the radicand must be greater than or equal to 0
37. Using interval notation, find the domain of each function.
a. b. c. d.
e. f. g. h.
38. Using interval notation, find the domain and range of each function. Use your graphing calculator if
necessary.
a. b. c.
d. e. f.
HW: p. 27 #s 11-15odd, 21-27odd
PIECEWISE FUNCTIONS
39. Sketch the graph of each piecewise function.
a. b.
40. Write the equations of each piecewise function, including restrictions. Assume scale = 1.
a. b.
a. b.
41. Evaluate each piecewise function as indicated.
a. ,
b. ,
HW: p. 27 #s 17, 19
COMPOSITION OF FUNCTIONS
42. If and , find the following:
a. =b. =c. =
d. =e. =d. =
HW: p. 29 #52
TRIG RATIO DEFINITIONS (OLD SCHOOL)
43. Complete each ratio using the labels for the right triangle.
44. The relationship between the primary and secondary trig ratios is that they are______!
TRIG RATIO DEFINITIONS (NEW SCHOOL)
45. Complete each ratio using the labels for the right triangle.
46. Define QUANDRANTAL ANGLE:
MNEMONIC FOR SIGNS OF NON-QUADRANT I RATIOS
47. Devise a mnemonic device to assist you in memorizing the signs of the non-quadrant one ratios.
48. CONVERT FROM DEGREES TO RADIANS
49. CONVERT FROM RADIANS TO DEGREES
50. Unit Circle – Circle with Radius = 1
51. Complete each special right triangle.
52. Complete the chart below.
Degrees
/ 0 / 30 / 45 / 60 / 90Radians
Sin
Cos
Tan
Csc
Sec
Cot53. EVALUATE – DRAW A QUICK REFERNCE TRIANGLE OR MEMORIZE THE CHART!
a.
b.
c.
d.
e.
f.
aa.
bb.
cc.
dd.
ee.
ff.
aaa.
bbb.
ccc.
ddd.
eee.
fff.
54. SOLVE EACH TRIG EQUATION FOR x FROM [0, 2)
a.
b.
c.
d.
e.
f.
aa.
bb.
cc.
dd.
ee.
ff.
55. SOLVE EACH MULTIPLE ANGLE TRIG EQUATION FOR x FROM [0, 2)
a.
b.
c.
Trigonometry – Basic Identities
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Sum and Difference Identities
Double Angles Identities
Complex Fractions
Notes:
a. Denominators must be factored completely to obtain the LCD.
b. Choose an LCD that will cancel out all existing denominators.
c. Use grouping symbols to keep your work organized.
d. Cancel out only factors (not terms) which are common to both the numerator and the denominator.
e. Recall, (m-n)/(n-m) = -1
For questions 1-2, determine the LCD only – DO NOT SIMPLIFY!
1.
2.
For questions 3-5, simplify.
3.
4.
5. -1
6.
7.
8.
9.
10.
11.
12.
13.
1