function intercepts interval notation domain and range equation of a line radian and angle measure inverse trig function circle and ellipse logarithm factoring

Formula Sheet Reciprocal Identities:

cscx

Quotient Identities:

tan x =

Pythagorean Identities:

sin2 x + cos2 x = I

Double Angle Identities:

sin 2x = 2 sin x cos x 2 tan x tan 2x = 1 – tan2 x

Logarithms: y = loge x is equivalent to

sin x sin x cos x

sec x =

1 cos x

cot x =

cos x sin x

cot x =

tan 2 x + I = sec2 x

tan x

I + cot2 x = csc2 x

cos 2x = cos2 x – sin2 x = I –2sin2 x = 2 cos2 x – I The Zero Exponent: x°=1, for x not equal to 0.

x = aY Multiplying Powers

Product property:

log,, mn = log, m + logb n

Quotient property:

m log,, — = log, in – log, n

Multiplying Two Powers of the Same Base:

(xa)(xb)

x(a+b)

Multiplying Powers of Different Bases:

(Xy)a = (Xa)(ya)

Power property:

log,, m'' = p log,, in

Property of equality: then m n

If log, in = log,, n,

Dividing Powers Dividing Two Powers of the Same Base:

( ia)/(xb) =

Dividing Powers of Different Bases:

(X/y)a = (xa)/(ya)

Change of base formula:

• log, n loge n = logb a

Fractional exponent:

=

Negative Exponents:

=)1Ixii

Slope-intercept form: y = mx + b

2

Point-slope form:

y = m(x – x, )+

Standard form:

Ax + By + C --• 0

Complex Fractions When simplifying complex fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the complex fraction. Example:

—7 6 x +1 5 x +1

7

6 x +1. x +1 5 • x +1 x +1

—7x-7-6 5

—7x-13 5

—2 3x —2 3x + + x x-4 = x x-4 . x(x— 4) = —2(x— 4) + 3x(x) 1 1 x(x— 4) 5(x)(x— 4)-1(x) 5 5 x4 x- 4

-2x + 8+ 3x2 5x2 —20x— x

=

3x2 — 2x+ 8 5x2 —21x

Simplify each of the following.

25 --a 1.

a

5±a

4 x+ 2 2. 10 5+ x+ 2

12 2x-3 3. 15 5+ 2x —3

2-

4

2x 3x-4 5. 32 x+ 3x— 4 1—

3

Function To evaluate a function for a given value, simply plug the value into the function for x. Recall: (f g)(x) = f (g(x)) OR f[g(x)] read "f of g of x" Means to plug the inside function (in this case g(x) ) in for x in the outside function (in this case, f(x)). Example: Given f (x) =2 x2 +1 and g(x) = x — 4 find f(g(x)). f(g(x))= f (x —4)

= 2(x — 4)2 + 1 = 2(x2 — 8x +16) + 1 =2x2 -16x+32+1 f (g(x)) = 2x2 —16x +33

Let f (x) = 2x +1 and g(x) = 2x2 —1. Find each. 8. f (t +1) =

6. f (2) =

7. g(-3) =

9. f[g(-2)]=

10. g[f (m + 2)].

Let 12. f

11

f

f (x)= sinx Find each exactly. Tr

)=

13.

3

Let f (x) = x2, g(x) = 2x + 5, and h(x) = x2 —1. Find each. 14. h[f(-2)]=

15. f[g(x —1)]=

4

16. g[h(x3)]=

f (x)

Systems

Use substitution or elimination method to solve the system of equations. Example: x2 +y-16x+39= 0 X2 -

y2 — 9 = 0

Elimination Method

Substitution Method

2x2 —16x + 30 = 0

Solve one equation for one variable.

x2 -8x+15 = 0 (x— 3)(x — 5) = 0 x=3andx=5 Plug x=3 and x= 5 into one original 52_ y2 _ 9 0 32 _ y2 9_ 0 —y2 = 0 16= y2 y = ±4 y=0 Points of Intersection (5,4), (5,-4-) and (3,0)

y2 = —x2 +16x — 39

(1st equation solved for y)

x2 — (—x2 +16x — 39)— 9 = 0 Plug what y2 is equal to into second equation. 2x2 —16x+ 30 = x 2 8x+15 = 0

(The rest is the same as previous example)

(x —3)(x— 5) = 0 x= 3 or x-5

Find the point(s) of intersection of the graphs for the given equations.

23.

x+y= 8 4x — y =7

24.

x2 + y =6 x+y=4

25.

x2 — 4y2 — 20x — 64y —172 = 0 16x2 +4y2 — 320x + 64y +1600 = 0

Interval Notation

26. Complete the table with the appropriate notation or graph. Solution

—2 < x

Interval Notation

Graph

4 [-1,7)

--/8

Also, recall that to PROVE one function is an inverse of another function, you need to show that: f (g(x)) = g(f (x)) = x Example: x–9 If: f (x) = 4 and g (x) = 4x + 9 show f(x) and g(x) are inverses of each other.

g(f (x)) = 4

(x-0 +9 4

=x-9+9 =

f (g(x)) =

(4x + 9) – 9

4 4x +9 –9 4 = 4x 4 -= x f (g(x)) = g(f (x)) = x therefore they are inverses of each other.

Provef and g are inverses of each other. x3 36. f (x) = — 2

g(x) = VZX

37. f (x) = 9 – x2 , x

0

g(x) =

x

Radian and Degree Measure 180° to get rid of radians and gradians convert to degrees.

gradians to get rid of degrees and 180° convert to radians.

Use

57c

Use

L

47r

46. Convert to degrees:

a. — 6

▪.

47. Convert to radians:

a. 45'

b. —17

c. 2.63 radians

5

c. 237

Angles in Standard Position 48. Sketch the angle in standard position. 5TC C. --

b. 230°

3

10

d. 1.8 radians

Reference Triangles 49. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. 2 a. — 7C 3

b. 225

7C

d. 30

C. --

4

Unit Circle You can determine the sine or cosine of a quadrantal angle by using the unit circle. The x-coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. Example:- sin 900 =1

50.

cos— = u 2

a.) sin180°

b.) cos 270°

c.) sin(-90°)

d.) sing'

e.) cos 360°

f.) cos(—r) 11

Graphing Trig Functions

y = sin x and y = cos x have a period of 2 r and an amplitude of I. Use the parent graphs above to help you — = period, sketch a graph of the functions below. For f (x). A sin(Bx + C)+ K, A = amplitude, 27c

c — = phase shift (positive C/B shift left, negative C/B shift right) and K = vertical shift. Graph two complete periods of the function. 51.

f(x) = 5 sinx

53. f (x) = cosi

52. f (x) = sin 2x

ir 4

54. f (x) = cos x — 3

Trigonometric Eq_uations: Solve each of the equations for 0 5_ x < 2r. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0 5_ x < 22r. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at the beginning of the packet.) 1

55. sinx = -2

56. 2cosx = Nrj.

12

57. cos 2x =

1 N/2

58. sin2 x =

2

59. sin 2x = -2

60. 2 cos2 x —1— cos x = 0

61. 4cos2 x — 3 = 0

62. sin2 x + cos 2x — cos x = 0

Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsin (x)

sin-1 (x)

irivers.e trig fimations are defined only in the quadrants as indicated below clue to their restricted dornainS. cos-lx 0 cos-1 x >0 tan-1 x >0

41 sin-1 x