Polynomial, Radical, and Rational Functions Lesson One: Polynomial Functions. ; polynomial degree is n; constant term is a 0

Transcription

1 Polynomial, Radical, and Rational Functions Lesson One: Polynomial Functions Answer Key Example : Leading coefficient is a n ; polynomial degree is n; constant term is a. i) ; ; - ii) ; ; - iii) ; ; i) Y ii) N iii) Y iv) N v) Y vi) N vii) N viii) Y ix) N Example : i) Even-degree polynomials with a positive leading coefficient have a trendline that matches an upright parabola. End behaviour: The graph starts in the upper-left quadrant (II) and ends in the upper-right quadrant (I). ii) Even-degree polynomials with a negative leading coefficient have a trendline that matches an upside-down parabola. End behaviour: The graph starts in the lower-left quadrant (III) and ends in the lower-right quadrant (IV). i) Odd-degree polynomials with a positive leading coefficient have a trendline matching the line y = x. The end behaviour is that the graph starts in the lower-left quadrant (III) and ends in the upper-right quadrant (I). ii) Odd-degree polynomials with a negative leading coefficient have a trendline matching the line y = -x. The end behaviour is that the graph starts in the upper-left quadrant (II) and ends in the lower-right quadrant (IV). Example : Zero of a Polynomial Function: Any value of x that satisfies the equation P(x) = is called a zero of the polynomial. A polynomial can have several unique zeros, duplicate zeros, or no real zeros. i) Yes; P(-) = ii) No; P(). Zeros: -,. The x-intercepts of the polynomial s graph are - and. These are the same as the zeros of the polynomial. "Zero" describes a property of a function; "Root" describes a property of an equation; and "x-intercept" describes a property of a graph. Example : Multiplicity of a Zero: The multiplicity of a zero (or root) is how many times the root appears as a solution. Zeros give an indication as to how the graph will behave near the x-intercept corresponding to the root. Zeros: - (multiplicity ) and (multiplicity ). Zero: (multiplicity ). Example a Zero: (multiplicity ). e) Zeros: - (multiplicity ) and (multiplicity ). Example b Example : i) Zeros: - (multiplicity ) and (multiplicity ). ii) y-intercept: (, -7.). iii) End behaviour: graph starts in QII, ends in QI. iv) Other points: parabola vertex (, -8). i) Zeros: - (multiplicity ) and (multiplicity ). ii) y-intercept: (, ). iii) End behaviour: graph starts in QII, ends in QIV. iv) Other points: (-, ), (-.67, -.), (, -). Example 6: i) Zeros: - (multiplicity ) and (multiplicity ). ii) y-intercept: (, ). iii) End behaviour: graph starts in QII, ends in QI. iv) Other points: (-, 6), (-.,.6), (, 6). i) Zeros: - (multiplicity ), (multiplicity ), and (multiplicity ). ii) y-intercept: (, ). iii) End behaviour: graph starts in QII, ends in QI. iv) Other points: (-, ), (-., -.), (., 8.), (, 9). Example 7: i) Zeros: -. (multiplicity ) and. (multiplicity ). ii) y-intercept: (, ). iii) End behaviour: graph starts in QIII, ends in QIV. iv) Other points: parabola vertex (, ). i) Zeros: -.67 (multiplicity ), (multiplicity ), and.7 (multiplicity ). ii) y-intercept: (, ). iii) End behaviour: graph starts in QIII, ends in QI. iv) Other points: (-, -7), (-.,.), (., -.8), (, ). Example 8: Example 9: Example : Example 6a Example 7a Example 7b Example 6b Example : x: [-,, ], y: [-69, 87, ] x: [-, 7, ], y: [-9, 78, ] x: [-,, ], y: [-6,, ]

2 Example : Example : Example : Example : V(x) = x( - x)(6 - x) < x < 8 or (, 8) Window Settings: x: [,8, ], y: [,, ] When the side length of a corner square is.9 cm, the volume of the box will be maximized at. cm. e) The volume of the box is greater than cm when.7 < x <.9. or (.7,.9) P product (x) = x (x + ); P sum (x) = x + x + x - x - =. Window Settings: x: [-,, ], y: [-, 76, ] Quinn and Ralph are since x =. Audrey is two years older, so she is. Window Settings: x: [, 6, ], y: [-.,.7, ] At. seconds, the maximum volume of.7 L is inhaled One breath takes. seconds to complete. 6% of the breath is spent inhaling. Example 6: Polynomial, Radical, and Rational Functions Lesson Two: Polynomial Division Example : Quotient: x - ; R = P(x): x + x - x - 6; D(x) = x + ; Q(x) = x - ; R = L.S.= R.S. Q(x) = x - + /(x + ) e) Q(x) = x - + /(x + ) Example : x - 7x /(x + ) x + x + x - x + - 9/(x + ) Example : x + x + - /(x - ) x - x + x - x + x - x - - /(x - ) Example : x - x x + x + + 6/(x - ) Example : a = - a = - Example 6: The dimensions of the base are x + and x - Example 7: f(x) = (x + )(x - ) g(x) = x + Q(x) = (x ) Example 8: f(x) = x - 7x - g(x) = x - Example 9: R = - R = -. The point (, -) exists on the graph. The remainder is just the y-value of the graph when x =. Both synthetic division and the remainder theorem return a result of - for the remainder. i) R = ii) R = - iii) R = - e) When the polynomial P(x) is divided by x - a, the remainder is P(. Example : P(-), so x + is not a factor. P(-), so x + is not a factor. P(/) =, so x - is a factor. P(-/), so x + is not a factor. Example : k = k = -7 k = -7 k = - Example : m = and n = -7 Example : m = and n = - Example : a = Example 9 (, -) (, ) Example 7c Interval Notation Math - students are expected to know that domain and range can be expressed using interval notation. () - Round Brackets: Exclude point from interval. [] - Square Brackets: Include point in interval. Infinity always gets a round bracket. Examples: x - becomes [-, ); < x becomes (, ]; x ε R becomes (-, ); -8 x < or x < becomes [-8, ) U [, ), where U means or, or union of sets; x ε R, x becomes (-, ) U (, ); - x, x becomes [-, ) U (, ]. Example : R = R =. The point (, ) exists on the graph. The remainder is just the y-value of the graph. Both synthetic division and the remainder theorem return a result of for the remainder. If P(x) is divided by x - a, and P( =, then x - a is a factor of P(x). e) When we use the remainder theorem, the result can be any real number. If we use the remainder theorem and get a result of zero, the factor theorem gives us one additional piece of information - the divisor fits evenly into the polynomial and is therefore a factor of the polynomial. Put simply, we're always using the remainder theorem, but in the special case of R = we get extra information from the factor theorem. Example Remainder Theorem (R = any number) Factor Theorem (R = )

3 Polynomial, Radical, and Rational Functions Lesson Three: Polynomial Factoring Example : The integral factors of the constant term of a polynomial are potential zeros of the polynomial. Potential zeros of the polynomial are ± and ±. The zeros of P(x) are - and since P(-) = and P() = The x-intercepts match the zeros of the polynomial e) P(x) = (x + )(x - ). Answer Key Example : P(x) = (x + )(x + )(x - ). All of the factors can be found using the graph. Factor by grouping. Example : P(x) = (x + )(x - ). Not all of the factors can be found using the graph. Factor by grouping. Example : P(x) = (x + )(x ). All of the factors can be found using the graph. No. Example : P(x) = (x + x + )(x - ). Not all of the factors can be found using the graph. x - 8 is a difference of cubes Example 6: P(x) = (x + x + )(x - ). Not all of the factors can be found using the graph. No. Example 7: P(x) = (x + )(x - )(x + ). Not all of the factors can be found using the graph. x 6 is a difference of squares. Example 8: P(x) = (x + )(x - ) (x - ). All of the factors can be found using the graph. No. Example 9: P(x) = /x (x + )(x ). Example : Width = cm; Height = 7 cm; Length = cm Example : -8; -7; -6 P(x) = (x + ) (x - ). Example : k = ; P(x) = (x + )(x - )(x - 6) Example : a = - and b = - Example : x = -,, and Quadratic Formula From Math -: The roots of a quadratic equation with the form ax + bx + c = can be found with the quadratic formula:

4 Polynomial, Radical, and Rational Functions Lesson Four: Radical Functions Example : x f(x) Domain: x ; Range: y - undefined Interval Notation: Domain: [, ); Range: [, ) 9 Example : Example : Example : Example : Example 6: Example 7: ORIGINAL: Domain: x ε R or (-, ) Range: y ε R or (-, ) ORIGINAL: Domain: x ε R or (-, ) Range: y 9 or (-, 9] ORIGINAL: Domain: x ε R or (-, ) Range: y - or [-, ) ORIGINAL: Domain: x ε R or (-, ) Range: y or [, ) TRANSFORMED: Domain: x - or [-, ) Range: y or [, ) TRANSFORMED: Domain: - x or [-, ] Range: y or [, ] TRANSFORMED: Domain: x or x 7 or (-, ] U [7, ) Range: y or [, ) TRANSFORMED: Domain: x ε R or (-, ) Range: y or [, )

5 Example 8: ORIGINAL: Domain: x ε R or (-, ) Range: y or (-, ] TRANSFORMED: Domain: x = - Range: y = ORIGINAL: Domain: x ε R or (-, ) Range: y. or [., ) TRANSFORMED: Domain: x ε R or (-, ) Range: y. or [., ) Example : Domain: d ; Range: h( or Domain: [, ]; Range: [, ]. When d =, the ladder is vertical. When d =, the ladder is horizontal. m h( Example 9: x = 7 (7, ) (7, ) d Example : Example : x = (, ) (, ) original time / original time h t t. Example : x = -,... (-, ) (, ) (-, ) (, ) h Example 6: Example : No Solution No Solution V(r) 6 No Solution Example : 6 r

6 Example : Polynomial, Radical, and Rational Functions Lesson Five: Rational Functions I x y undef... The vertical asymptote of the reciprocal graph occurs at the x-intercept of y = x..the invariant points (points that are identical on both graphs) occur when y = ±.. When the graph of y = x is below the x-axis, so is the reciprocal graph. When the graph of y = x is above the x-axis, so is the reciprocal graph. Example : Original Graph: Domain: x ε R or (-, ); Range: y ε R or (-, ) Reciprocal Graph: Domain: x ε R, x or (-, ) U (, ); Range: y ε R, y or (-, ) U (, ) Asymptote Equation(s): Vertical: x = ; Horizontal: y = Original Graph: Domain: x ε R or (-, ); Range: y ε R or (-, ) Reciprocal Graph: Domain: x ε R, x or (-, ) U (, ); Range: y ε R, y or (-, ) U (, ) Asymptote Equation(s): Vertical: x = ; Horizontal: y = Example : x y x y undef x y undef...9. Example : Original: x ε R; y - or D: (-, ); R: [-, ). Reciprocal: x ε R, x -, ; y - or y > or D: (-, -) U (-, ) U (, ); R: (-, -] U (, ) Asymptotes: x = ±; y = Original: x ε R; y / or D: (-, ); R: (-, /]. Reciprocal: x ε R, x -, ; y < or y or D: (-, -) U (-, ) U (, ); R: (-, ) U [, ) Asymptotes: x = -, x = ; y =. The vertical asymptotes of the reciprocal graph occur at the x-intercepts of y = x -.. The invariant points (points that are identical in both graphs) occur when y = ±.. When the graph of y = x - is below the x-axis, so is the reciprocal graph. When the graph of y = x - is above the x-axis, so is the reciprocal graph. Original: x ε R; y - or D: (-, ); R: [-, ). Reciprocal: x ε R, x, 8; y -/ or y > or D: (-, ) U (, 8) U (8, ); R: (-, -/] U (, ) Asymptotes: x =, x = 8; y = Original: x ε R; y or D: (-, ); R: [, ). Reciprocal: x ε R, x ; y > or D: (-, ) U (, ); R: (, ) Asymptotes: x = ; y =

7 Example (continue: e) Original: x ε R; y or D: (-, ); R: [, ). Reciprocal: x ε R; < y / or D: (-, ); R: (, /] Asymptotes: y = f) Original: x ε R; y -/ or D: (-, ); R: (-, -/]. Reciprocal: x ε R; - y < or D: (-, ); R: [-, ) Asymptotes: y = Example : Example 6: x =.; y = x = -, 6; y = x = -.,,. ; y = y = Example 7: VS: VT: down VS: ; HT: left VS: ; HT: right; VT: up Example 8: VT: down HT: right; VT: up VS: ; HT: right; VT: down VS: ; HT: right; VT: down Example 9: Example : Illuminance V.S. Distance for a Fluorescent Bulb I P(V) = nrt(/v). / original original 8. kpa L/mol K e) See table & graph f) See table & graph V (L)..... P (kp... P Pressure V.S. Volume of. mol of a gas at 7. K V / original /9 original original 6 original e) See table & graph f) See table & graph d (m) ORIGINAL 8 I (W/m ) d

8 Polynomial, Radical, and Rational Functions Lesson Six: Rational Functions II Example : y = x x - 9 x + y = x + x + y = x - 6 x - x - y = x - x - x Example : y = x x - y = x x - y = x x + 9 y = x - x - 8 x - x - 6 Example : y = x + x + x + y = x - x + x - y = x + x - y = x - x - 6 x + Example : Example : Example 6: Example 7: i) Horizontal Asymptote: y = ii) Vertical Asymptote(s): x = ± iii) y - intercept: (, ) iv) x - intercept(s): (, ) v) Domain: x ε R, x ±; Range: y ε R i) Horizontal Asymptote: y = ii) Vertical Asymptote(s): x = - iii) y - intercept: (, -) iv) x - intercept(s): (, ) v) Domain: x ε R, x -; Range: y ε R, y i) Horizontal Asymptote: None ii) Vertical Asymptote(s): x = iii) y - intercept: (, 8) iv) x - intercept(s): (-, ), (, ) v) Domain: x ε R, x ; Range: y ε R vi) Oblique Asymptote: y = x + i) y = x - ii) Hole: (, -) iii) y - intercept: (, -) iv) x - intercept(s): (, ) v) Domain: x ε R, x ; Range: y ε R, y - or D: (-, ) U (, ); R: (-, ) or D: (-, -) U (-, ) U (, ); R: (-, ) or D: (-, -) U (-, ); R: (-, ) U (, ) or D: (-, ) U (, ); R: (-, -) U (-, )

9 Example 8: Example : d s t Equal times. Cynthia Alan x + x x + x Cynthia: 9 km/h; Alan: 6 km/h Graphing Solution: x-intercept method. Example 9: (6, ) Example : Example : x = d s t Sum of times equals h. (, ) Upstream Downstream x - x + x - x + (, ) Canoe speed: km/h Graphing Solution: x-intercept method. Example : x = -/ and x = (-., ) (, ) (-., ) (, ) Example : Example 6: (-., -6) (, -6) Example : x =. x = is an extraneous root Number of goals required: 6 Graphing Solution: x-intercept method. Mass of almonds required: 9 g Graphing Solution: x-intercept method.. (, -) (, ) (6, ) - (9, )

10 Transformations and Operations Lesson One: Basic Transformations Example : Example : Example : Example : Example : Example 6:

11 Example 7: Example 8: Example 9: Example : Example : or Example : R(n) = n C(n) = n + loaves C (n) = n + R (n) = 6n e) loaves $ Example a R(n) C(n) (, ) $ (, ) Example e R (n) C (n) Example : 6 8 n 6 8 n Example : metres or

12 Transformations and Operations Lesson Two: Combined Transformations Example : a is the vertical stretch factor. b is the reciprocal of the horizontal stretch factor. h is the horizontal displacement. k is the vertical displacement. i. V.S. / H.S. / ii. V.S. H.S. iii. V.S. / H.S. Reflection about x-axis iv. V.S. H.S. / Reflection about x-axis Reflection about y-axis Example : Example : Example : H.T. left i. H.T. right ii. H.T. left V.T. up V.T. down iii. H.T. right V.T. down iv. H.T. 7 left V.T. up Example : Stretches and reflections should be applied first, in any order. Translations should be applied last, in any order. i. V.S. H.T. left V.T. up ii. H.S. Reflection about x-axis V.T. down iii. V.S. / Reflection about y-axis H.T. left; V.T. down iv. V.S. ; H.S. / Reflection about x-axis Reflection about y-axis H.T. right; V.T. up Example 6: Example 7: Example 8: (, ) (, 6) m = 8 and n = Example 9: y = -f(x ) y = -f[(x + )] Example : Axis-Independence Apply all the vertical transformations together and apply all the horizontal transformations together, in either order. Example : H.T. 8 right; V.T. 7 up Reflection about x-axis; H.T. left; V.T. 6 down H.S. ; H.T. left; V.T. 7 up H.S. /; Reflection about x & y-axis; H.T. right; V.T. 7 down. e) The spaceship is not a function, and it must be translated in a specific order to avoid the asteroids.

13 Transformations and Operations Lesson Three: Inverses Example : Line of Symmetry: y = x Example : Example : Example : Restrict the domain of the original function to - x - or - x Restrict the domain of the original function to x or x. Original: D: x ε R Inverse: D: x ε R The inverse is a function. Original: D: x ε R Inverse: D: x ε R The inverse is a function. Example : Example 6: Restrict the domain of the original function to x or x Restrict the domain of the original function to x - or x -. D: x ε R D: x Example 7: (, 8) True. f - ( = a f() = k = Example 8: 8 C is equivalent to 8. F F is equivalent to 7.8 C C(F) can't be graphed since its dependent variable is C, but the dependent variable on the graph's y-axis is F. This is a mismatch. e) f) The invariant point occurs when the temperature in degrees Fahrenheit is equal to the temperature in degrees Celsius. - F is equal to - C. F - - C (-, -) - - F(C) F - (C)

14 Transformations and Operations Lesson Four: Function Operations Example : x (f + g)(x) f(x) Domain: -8 x or [-8, ] Range: -9 y or [-9, ] x (f - g)(x) DNE 9 f(x) Domain: - x ; or [-, ] Range: y or [, ] -9 g(x) g(x) -9 6 DNE Domain: x (f g)(x) -6 x x f(x) or [-6, ] Range: g(x) -8 y - or [-8, -] - - (f g)(x) DNE f(x) g(x) Domain: - x or [-, ] Range: -8 y - or [-8, -] 6 DNE - Example : i. (f + g)(-) = - ii. h(x) = -; h(-) = - i. (f g)(6) = 8 ii. h(x) = x ; h(6) = 8 i. (fg)(-) = -8 ii. h(x) = -x + x - ; h(-) = -8 i. (f/g)() = -. ii. h(x) = (x - )/(-x + ); h() = DNE Reminder: Math - students are expected to know that domain and range can be expressed using interval notation. Example : g(x) f(x) f(x) g(x) f(x) g(x) g(x) f(x) m(x) Domain: < x or (, ] Range: < y or (, ] Example : Domain: x - or [-, ) Range: y 9 or (-, 9] Domain: < x or (, ] Range: - y or [-, ] Example : Domain: x > - or (-, ) Range: y > or (, ) f(x) f(x) g(x) g(x) g(x) g(x) f(x) f(x) Domain: x - or [-, ) Range: y or [, ) Transformation: y = f(x) - Domain: x - or [-, ) Range: y - or (-, -] Transformation: y = -f(x). Domain: x ε R or (-, ) Range: y -6 or (-, -6] Transformation: y = f(x) - Domain: x ε R or (-, ) Range: y - or (-, -] Transformation: y = /f(x)

15 Example 6: g(x) f(x) g(x) f(x) g(x) f(x) g(x) f(x) Domain: x ε R, x ; Range: y ε R, y Domain: x ε R, x ; Range: y ε R, y Domain: x ε R, x -; Range: y ε R, y Domain: x -, x -; Range: y ε R, y or D: (-, ) U (, ); R: (-, ) U (, ) or D: (-, ) U (, ); R: (-, ) U (, ) or D: (-, -) U (-, ); R: (-, ) U (, ) or D: [-, -) U (-, ); R: (-, ) U (, ) Example 7: A L (x) = 8x 8x A S (x) = x x A L (x) - A S (x) = ; x = A L () + A S () = ; e) The large lot is.67 times larger than the small lot Example 8: $ R(n) Example 9: R(n) = n; E(n) = n + 6; P(n) = 8n 6 When games are sold, the profit is $6 Greg will break even when he sells games - 6 E(n) P(n) n The surface area and volume formulae have two variables, so they may not be written as single-variable functions. e) f) Transformations and Operations Lesson Five: Function Composition Example : Order matters in a f) x g(x) f(g(x)) x f(x) g(f(x)) composition of functions m(x) = x e) n(x) = (x ) m(x) n(x) Example : m() = n() = - p() = - q(-) = -6 Example : m(x) = x n(x) = x 6 p(x) = x 6x + 6 q(x) = x e) All of the results match Example : m(x) = (x + ) n(x) = (x + ) The graph of f(x) is horizontally stretched by a scale factor of /. The graph of f(x) is vertically stretched by a scale factor of.

16 Example : Domain: x 8 Domain: x Example 6: h( x ) = x + Domain: x ε R, x - h( x ) = x + + Domain: x - Example 7: Domain: x ε R, x - Domain: x - Example 8: e) f) Example 9: Example : Example : Example : Example : (f - f)(x) = x, so the functions are inverses of each other. (f - f)(x) x, so the functions are NOT inverses of each other. The cost of the trip is $.. It took two separate calculations to find the answer. V( =.8d M(V) =.V M( =.8d e) Using function composition, we were able to solve the problem with one calculation instead of two. A(t) = 9t A = 8 cm t = 7 s; r = cm a( =.c j( = 78.7a b( =.678a b( =.667c h = cm M( 6 6 d

17 Exponential and Logarithmic Functions Lesson One: Exponential Functions Answer Key Example : Parts (a-: Domain: x ε R or (-, ) Range: y > or (, ) x-intercept: None y-intercept: (, ) Asymptote: y = An exponential function is defined as y = b x, where b > and b. When b >, we get exponential growth. When < b <, we get exponential decay. Other b-values, such as -,, and, will not form exponential functions. Example : ; ; ; ; Example : Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Example : Domain: x ε R or (-, ) Range: y > - or (-, ) Asymptote: y = - Domain: x ε R or (-, ) Range: y > - or (-, ) Asymptote: y = - Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Domain: x ε R or (-, ) Range: y > or (, ) Asymptote: y = Example : Example 6: e) V.S. of 9 f) equals H.T. units left. See Video

18 Example 7: Example 8: Example 9: Example : Example : Example : Example : Example : e) f) infinite solutions no solution Example : m Example c m Example d 8 g See Graph. 9 years (9,.) t 6 t Example 6: bacteria See Graph 6 hours ago B Example 6c 8 6 (-6, ) B Example 6d Watch Out! The graph requires hours on the t-axis, so we can rewrite the exponential function as: t -8 t Example 7: ; 69 MHz ; $6 Example 8: 8,7 years 6 77 years P % (.7, ) Example 8b P % Example 8d (76.7,.) 6 8 t 6 8 t Example 9: A( t ) = (.) t $6.7 Interest: $6.7 See graph $ Example 9c P% (8, ) Example 9d 8 years e) $66.; $66.; $66.7 As the compounding frequency increases, there is less and less of a monetary increase. t t

19 Exponential and Logarithmic Functions Lesson Two: Laws of Logarithms Answer Key Example : Example : Example : Example : Example 6: The base of the logarithm is b, a is called the argument of the logarithm, and E is the result of the logarithm. In the exponential form, a is the result, b is the base, and E is the exponent. i. ; ; ; ii. ; ; ; i. log ii. Example : e) f) g) h) e) f) g) e) f) g) h) e) f) g) h) h) Example : Example : Example 7: Example 8: Example 9: Example : e) e) e) e) f) f) f) f) Example : Example : Example : g) g) g) g) h) h) h) h) ± Example 6: Example 7: Example 8: Example 9: Example : e) f) g) h) e) f) g) h) e) f) g) see video h) e) f) g) h) e) f) g) h)

20 Example : Exponential and Logarithmic Functions Lesson Three: Logarithmic Functions See Graph See Graph See Video e) i) -, y = x y = log x f(x) = x ii), iii), Domain x ε R x > iv).8 f - (x) = log x Range y > y ε R - - x-intercept y-intercept Asymptote Equation none (, ) y = (, ) none x = f) y = log x, y = log x, and y = log - x are not functions. is a function. g) The logarithmic function y = log b x is the inverse of the exponential function y = b x. It is defined for all real numbers such that b> and x>. h) Graph log x using logx/log Example : D: x > D: x > D: x > or (, ) or (, ) or (, ) D: x > or (, ) or (-, ) or (-, ) or (-, ) or (-, ) A: x = A: x = A: x = A: x = Example : D: x > D: x > D: x > or (, ) or (, ) or (, ) D: x > or (, ) or (-, ) or (-, ) or (-, ) or (-, ) A: x = A: x = A: x = A: x = Example : D: x > D: x > - D: x > or (, ) or (-, ) or (, ) D: x > - or (-, ) or (-, ) or (-, ) or (-, ) or (-, ) A: x = A: x = - A: x = A: x = - Example : D: x > - or (-, ) D: x > or (, ) D: x > - or (-, ) D: x > - or (-, ) or (-, ) or (-, ) or (-, ) or (-, ) A: x = - A: x = A: x = - A: x = -

21 Example 6: D: x > or (, ) D: x > or (, ) D: x > or (, ) D: x > or (, ) or (-, ) or (-, ) R: y > log or (log, ) or (-, ) A: x = A: x = A: none A: x = Example 7: x = 8 No Solution (8, 6) No Solution (-.6,.7) Example 8: x = 8 x = (, ) x = (8, ) (, ) Example 9: Example : Example : (y-axis) (-, ) and (, ) e) e) e) Example : Example : Example : Example : 6 db ph = cents. m. W/m - mol/l 78 Hz.6 times stronger See Video times more intense See Video times stronger See Video cents separate the two notes e) times stronger e) times more intense e) ph = f) See Video f) See Video f) ph = g). g) db g) times more acidic h). h) 7 db

22 Example : Trigonometry Lesson One: Degrees and Radians The rotation angle between the initial arm and the terminal arm is called the standard position angle. An angle is positive if we rotate the terminal arm counterclockwise, and negative if rotated clockwise. The angle formed between the terminal arm and the x-axis is called the reference angle. If the terminal arm is rotated by a multiple of 6 in either direction, it will return to its original position. These angles are called co-terminal angles. e) A principal angle is an angle that exists between and 6. 6 Note: For illustrative purposes, all diagram angles will be in degrees. f) The general form of co-terminal angles is c = p + n(6 ) using degrees, or c = p + n() using radians.,, 76,, 8, -, -67, -, -9 - Example : i. One degree is defined as /6 th of a full rotation. ii. One radian is the angle formed when the terminal arm swipes out an arc that has the same length as the terminal arm. One radian is approximately iii. One revolution is defined as 6º, or pi. It is one complete rotation around a circle. Conversion Multiplier Reference Chart degree radian revolution rev degree 8 6 radian revolution 8 6 rev rev rev i.. rad ii..6 rev iii iv.. rev v. 7 vi..7 rad i..79 rad ii. / rad Example :. rad 7/6 rad / rev. e) 7 f).7 rad g) / rev h) 8 i) 6 rad Example : = = = 9 = 6 = = = Example : r = r = 8 r = 6 (or.98 ra -6 r = (or / ra e) r = (or /7 ra = 8 = 6 = = = = = = = Example 6: p =, r = p =, r = p = 6, r = p =, r = 6 (or p =.7, r =.) (or p = /, r = /) Example 7: = 7 = 6, = 6 p = -9, = p c = -,, 78 c = -8, -,, Example 8: p = 9 p = 8 p = p = (or.8 ra (or / ra (or /6 ra 9 8 = 67, = p = 8, = p c = -, c = -96, -6, -,, (or c = -.78,.) 8, (or c = -6/, -/, -/, /, /, /) Example 9: c = 8 c = -8/ c = c = /

23 Example : p =.6, r = p =.69, r = Example : sin: QI: +, QII: +, QIII: -, QIV: - cos: QI +, QII: -, QIII: -, QIV: + tan: QI +, QII: -, QIII: +, QIV: - csc: QI: +, QII: +, QIII: -, QIV: - e) sec: QI +, QII: -, QIII: -, QIV: + f) cot: QI +, QII: -, QIII: +, QIV: - g) sin & csc share the same quadrant signs. cos & sec share the same quadrant signs. tan & cot share the same quadrant signs Example : i. QIII or QIV ii. QI or QIV iii. QI or QIII i. QI ii. QIV iii. QIII i. none ii. QIII iii. QI Example : p =.6, r =.6 p =.6, r =.8 (or p =. rad, r =.9 ra (or p =.7 rad, r =. ra Example : p =., r = 6.87 p = 6., r = Example : If the angle could exist in either quadrant or... I or II I or III I or IV II or III II or IV III or IV The calculator always picks quadrant I I I II IV IV Each answer is different because the calculator is unaware of which quadrant the triangle is in. The calculator assumes Mark s triangle is in QI, Jordan s triangle is in QII, and Dylan s triangle is in QIV. Example 6: The arc length can be found by multiplying the circumference by the sector percentage. This gives us: a = r / = r.. cm.9.6 cm e) n = 7/6 Example 7: The area of a sector can be found by multiplying the area of the full circle by the sector percentage to get the area of the sector. This gives us: a = r / = r /. 8/ cm cm 8/ cm e) cm Example 8: 6 /s.7 rad/s. km 7 rev/s e).6 rev/s Example 9: /7 rad/s 68. km

24 Trigonometry Lesson Two: The Unit Circle Example : i. ii. i. Yes ii. No (.6,.8) (.,.) i. ii. - - iii. y = iv. Example : See Video. Example : - e) f) g) h) Example : Example 6:,,,, Example 7: See Video. Example : e) - f) g) h),,,,,,,,,,,, Example : C = The central angle and arc length of the unit circle are equal to each other. a = / a = 7/6 Example : The unit circle and the line y = do not intersect, so it's impossible for sin to equal. Range Number Line cos & sin csc & sec tan & cot.,.7 e) y = Example 8: - undefined e) f) - g) h) Example 9: Example : Example : - undefined undefined Example 6: Inscribe a right triangle with side lengths of x, y, and a hypotenuse of into the unit circle. We use absolute values because technically, a triangle must have positive side lengths. Plug these side lengths into the Pythagorean Theorem to get x + y =. Use basic trigonometric ratios (SOHCAHTOA) to show that x = cos and y = sin. p = 67., r =.68 x y Example : See Video. Example : P(/) means "point coordinates at /". e) P() = (-.99,.) Example 7: (67, ) (-79, ) Example 8: See Video 6 m

25 Example : Trigonometry Lesson Three: Trigonometric Functions I (-/6, ), (-/6, -), (7/6, ) (-/, -), (/, 6), (7/, -8) (-6, 8), (-, -8), (, -) (-, ), (/, -), (/, -) Example : y = sin a = P = c = e) d = f) = n, nεi g) (, ) h) Domain: ε R, Range: - y y Example : y = cos a = P = c = e) d = f) = / + n, nεi g) (, ) h) Domain: ε R, Range: - y y Example : y = tan Tangent graphs do not have an amplitude. P = c = e) d = f) = n, nεi g) (, ) h) Domain: ε R, / + n, nεi, Range: y ε R y Example : Example 7: Example 6: Example 8: - Example 9:

26 Example : Example : Example : Example : Example : Example : Example 6:

27 Example 7: y = sec P = Domain: ε R, / + n, nεi; Range: y -, y = / + n, nεi y y Example 8: y = csc P = Domain: ε R, n, nεi; Range: y -, y = n, nεi y y Example 9: y = cot P = Domain: ε R, n, nεi; Range: yεr = n, nεi y y Example : Domain: ε R, / + n, nεi; (or: ε R, / ± n, nεw) Range: y -/, y / Domain: ε R, / + n/, nεi; (or: ε R, / ± n/, nεw) Range: y -, y Domain: ε R, / + n, nεi; (or: ε R, / ± n, nεw) Range: y -, y Domain: ε R, n(), nεi; (or: ε R, ±n(), nεw) Range: y ε R

28 Trigonometry Lesson Four: Trigonometric Functions II Example : Example : y y h y 8º 6º º 6 t 8 6 x Example : Example : y x y x Example 6: Example : e) y x y x Example 7: Example 8: The b-parameter is doubled when the period is halved. The a, c, and d parameters remain the same. The d-parameter decreases by units, giving us d =. All other parameters remain unchanged.

29 Mice Answer Key Example 9: Example : h(t) 7 Decimal daylight hours: 6.77 h,.8 h, 7.8 h,.8 h, 6.77 h d(n) 6 =.cos d ( n) ( n ).86 h e) 6 days t 8 If the wind turbine rotates counterclockwise, we still get the same graph. Example : h() - n Example : Decimal hours past midnight:. h, 8. h,. h,. h h(t) 6.7 m e).% 8 9, Example : h(t).. The angle of elevation increases quickly at first, but slows down as the helicopter reaches greater heights. The angle never actually reaches 9. Example : 8 6 t Population See Video. 6 M(t) Owls 8 O(t) Time (years).86 m t.6 s Example 6:. m h(t). (8,.). Example : h(t) 6 t 6 Example 7:.6 s and 8. s h(t) 9 t 7 7 (.6,.) (8.,.) 8. m 6.78 s 6 t

30 Trigonometry Lesson Five: Trigonometric Equations Note: n ε I for all general solutions. Example :,, Example : no solution e) f) Example : Example : Intersection point(s) of original equation -intercepts Intersection point(s) of original equation -intercepts

31 Example : Example 6: 97.6 and and and and. The unit circle is not useful for this question Example 7: - - Example 8: No Solution e) f)

32 Example 9: Example : 7 7 No Solution Intersection point(s) of original equation -intercepts Intersection point(s) of original equation -intercepts Example : Example : 6 6 The unit circle is not useful for this question Example : -

33 Example : Example : Example 6: Example 7: Example 8: Example 9: Example : Example : See Video Approximately days. Visible % See graph..66 rad (or 6.6 ) d t -8

34 Trigonometry Lesson Six: Trigonometric Identities I Note: n ε I for all general solutions. Example : Identity Equation i) ii) iii) iv) v) Not an Identity Not an Identity Identity Identity Not an Identity Example : Use basic trigonometry (SOHCAHTOA) to show that x = cos and y = sin. x y Verify that the L.S. = R.S. for each angle. The graphs of y = sin x + cos x and y = are the same. - Divide both sides of sin x + cos x = by sin x to get + cot x = csc x. Divide both sides of sin x + cos x = by cos x to get tan x + = sec x. Example : e) Verify that the L.S. = R.S. for each angle. f) The graphs of y = + cot x and y = csc x are the same Example :, The graphs of y = tan x + and y = sec x are the same Example :

35 Example 6: Example 7: Example 8:

36 Example 9: See Video Example : See Video Example : See Video Example : See Video Example : See Video Example : See Video The graphs are NOT identical. The R.S. has holes. The graphs are identical. The graphs are identical Example :,, ,, Example 6:,, - Note: All terms from the original equation were collected on the left side before graphing ,, - -

37 Example 7:,, Note: All terms from the original equation were collected on the left side before graphing Note: All terms from the original equation were collected on the left side before graphing ,, Example 8: Example 9: See Video

38 Trigonometry Lesson Seven: Trigonometric Identities II Note: n ε I for all general solutions. Example : e) f) Example : Example : See Video Example : Example : See Video Example 6: i. ii. iii. undefined (answers may vary) i. (answers may vary) i. ii. iii. iv. ii. iii. iv. Example : d =.6 and = 6. Example : At, the cannonball hits the ground as soon as it leaves the cannon, so the horizontal distance is m. At, the cannonball hits the ground at the maximum horizontal distance,. m. At 9, the cannonball goes straight up and down, landing on the cannon at a horizontal distance of m Examples 7 - : Proofs. See Video. Example : Example : A 9 The maximum area occurs when =. At this angle, the rectangle is the top half of a square. 9 i. ii. iii. Example : Example 6: Example 7: i. y = f() + g() i. 6 6 y = f() + g() Example 8: 7 Example 9: ii. The waves experience constructive interference. iii. The new sound will be louder than either original sound. All of the terms subtract out leaving y =, A flat line indicating no wave activity. -6 ii. The waves experience destructive interference. iii. The new sound will be quieter than either original sound. Example : See Video. Example : See Video.

39 Permutations and Combinations Lesson One: Permutations Example : Six words can be formed. = 6 P See Video e) P + P + P Example : (-)! Does not exist. e) f) g) n n h) n + n Example : 7! Example : 8 6 e) 6 f) Example : e) 676 Example 6: Example 7: 8 8 e) f) 6 Example 8: 68 Example 9: 7 Example : 6 e) 7 Example : P P or P Example : n = 6 n = n = n = Example : n = 8 r = n = n = Permutations and Combinations Lesson Two: Combinations Example : The order of the colors is not important. 6 C See Video e) C + C Example : ; 8 Example : Example : e) Example : Example 6: Example 7: Example 8: e) Example 9: 6 6 C e) C Example : n = 7 C n = n = 6 Example : n = All n-values n = n = Example : e) See Video f) 8 g) h) Example : 6!/(!) e) f) g) 98 h) Example : e) 9 f) 6 g) 9 h) 66 Example : e) f) g) h) Example 6: See Video e) n = 8 f) 6 g) See Video h) 6 Permutations and Combinations Lesson Three: The Binomial Theorem Example : The eighth row of Pascal's Triangle is:, 7,,,,, 7,. See Video. Note that rows and term positions use a zero-based index. There is symmetry in each row. For example, the second position of the sixth row is equal to the second-last position of the same row. Example : 8 C ; C C = k = and 8, so the fourth and ninth positions have a value of 6. Example : 66 Example : The binomial theorem states that a binomial power of the form (x + y) n can be expanded into a series of terms with the form n C k x n-k y k, where n is the exponent of the binomial (and also the zero-based row of Pascal's Triangle), and k is the zero-based term position. Example : Example 6: Example 7: Example 8: Example 9: Example :