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1: By Esther Afolayan and Ania Ryt | The ultimate how-to guide for grade 11 Academic Math

2: Contents: Functions How to determine functions? How to evaluate functions for a specific number? How to get the inverse function? How to transform parent functions? How to list points of a transformed functions? Polynomials How do I determine if two polynomials are equivalent? How to add and subtract polynomials? How to multiply polynomials? How to factor polynomials? Rational expressions How to simplify rational expressions? How do I write restrictions? How to determine the hole, vertical asymptote, zero and y-intercepts? How to multiply and divide rational expressions? How to add and subtract rational expressions? | ......................................................... 4 ............................. 5 ............................................................. 7 ........................ 7 ................... 9 .... 10 ...................................................... 11 ......................................................... 12 .............. 13 ............................. 13 ................................ 14 ...................................... 16 ................ 17 ................................ 17 .................... 18 ...................................................... 19 ..................................................... 20

3: Radicals How to simplify radicals? How to add and subtract radicals? How to multiply radicals? How to use fractions in exponents with rational expressions? Exponents How to multiply powers? How to divide powers? How to do a power of a power? Exponential functions How to apply the exponential function? Trigonometry How to do the reciprocals of trigonometric ratios? How to graph a sinusoidal function by hand? How to find the equation of a sinusoidal function? How to determine the number of triangles possible in an ambiguous case? Check your understanding Answers | ......................................................... 21 ................................ 22 ................... 22 ................................. 23 ....................................... 23 ...................................................... 25 ................................. 26 .................................... 26 ....................... 27 .................................. 28 ........... 29 ................................................. 30 .............................................................. 31 .............................................................. 31 .......................................................... 32 ..................... 33 .......................... 34 .......................................................... 38

4: Functions

5: There are 5 ways to determine if a relation is a function or not Mapping Check if more than one arrow goes from any of the x-values. Non Function Function Graphing Draw the graph of the relation and perform the vertical line test - draw a straight, vertical line on the graph. If it is a function you should be able to draw it anywhere and have it intersect the graph only once. Non Function Function | How do I determine functions? | 5

6: With a set of ordered pairs Check if any x-values are repeating, with a different y-value. Using a table of values Check if any of the x values are repeating, and have more than one y value. Non function Function Using an equation You must solve the equation for different x-values. If you can find a value of x that gives more than one value of y, the relation is not a function.

7: How to evaluate functions for a specific number? | How to get the inverse function? | There are two ways of doing the inverse of operation Switching x and y You have to switch up the y and x in the given equation, then isolate the y. | 7 | 7

8: Re-writing the equation backwards You have to write the equation in reverse order and with opposite actions that were done to the x. So for example, if in the original equation x was first multiplied by 3, you will DIVIDE the 3 LAST. | x 3 -5 | +5 / 3

9: How to transform parent functions? | the general formula for all parent functions is f(x)=a(kx-d)+c The variables in the formula are responsible for: a - vertical transformations and reflections k - horizontal transformations and reflections d - horizontal translations c - vertical translations | APPLYING TRANSFORMATIONS TO DIFFERENT PARENT FUNCTIONS | |a|>1 - vertical stretch by factor of a 0<|a|<1 - vertical compression by factor of a a<0 - reflection in the x-axis |k|>1 - horizontal compression by factor of k 0<|k|<1 - horizontal stretch by factor of k k<0 - reflection in the y-axis d>0 - d units to the right d<0 - units to the left c>0 - c units up c<0 - c units down | Linear Quadratic Square root Reciprocal Absolute value | 9

10: How to list points of transformed function? | The formula you will apply to the points is (x/k+d, ay+c). Therefore for example if you have a transformed equation of the quadratic function that is f(x)=3(2x+2) +8, the points would be (x/2+2, 3y+8). The table of it would transform this way: | 2 | It's sometimes easier to apply the transformations first, then the translations, but it can also be done in one step. | You can graph any functions by simply plotting some of the points and following the pattern.

11: Polynomials | 11

12: The polynomials ARE equivalent if: - Two different polynomials simplify to the same expression - Or they create the same graph | How can I determine if two polynomials are equivalent? | The polynomials ARE NOT equivalent if: - If you substitute numbers for the unknown variable and solve it, resulting in different answers *just because this method creates the same answer it does not mean they are equivalent, because they may create different answers for different numbers* | For example, if we were to find if is equivalent to | Method 1: Simplify Method 2: Evaluate

13: How to add and subtract polynomials? | You can determine the SUM of two or more polynomials by simply collecting like terms. | The DIFFERENCE is determined by adding the opposite of the polynomial, which means changing all the signs in the second equation and adding it to the first one. | How do I multiply polynomials? | You can figure out the product by using the distributive property (expanding), then collecting like terms. You have to multiply each term in the polynomial by each term in the second polynomial. | 13

14: How do I factor polynomials? | Factoring a polynomial is basically the opposite of expanding, therefore writing it as a product. So in the end you end up with the two brackets rather than the long bracket-less equation. | Expanding Factoring | The techniques you were taught in grade 10 are: - Common factoring - Simple trinomial - Complex trinomial - Difference of squares - Perfect square - And quadratic formula | But there are also other like: - Grouping - Decomposition - And the illegal move | Grouping | ( ) ( ) | n n 6 3 | n ( ) 3( ) | 1. group numbers into two 2. common factor out of both brackets 3. write the numbers from outside the bracket together, and inside the bracket as one

15: Decomposition | x | ( ) ( ) | x x | P: -30 S: 1 -5. 6 | - 5 + 6 | 1. Find numbers that add and multiply 2. Separate the x to two terms and multiply it by the numbers you found previously 3. Group into two and continue with the grouping method | Illegal move | P: -30 S: 1 -5. 6 | - 5 6 10x 10x | 10x | 1. Find numbers that add and multiply. 2. Write the numbers separately and divide each by the first term(without the squared) 3. Reduce 4. Write each fraction as a term | 15

16: Rational Expressions

17: How to simplify rational expressions? | 1. Factor the denominator and numerator separately 2. Find any terms that will cancel each other out between the numerator and denominator 3. Simplify, and write the | ( )( ) | You need to find a number, that if substituted for the unknown variable in the denominator, will make the whole denominator equal to zero, and that is your restriction. Therefore if a=3 (a-3) the bracket equals =(3-3) 0 x (a+3) = 0 zero because: =0 so 'a' cannot equal 3 | How do I write restrictions? | 17

18: How do I find the hole, vertical asymptote, zero and y-intercepts in a rational function? | = (x+7)(x-4) (x-4)(x+3) | x +3x-28 x -x-12 | 2 2 | 0 +0x-28 0-0-12 | 2 2 | -28 -12 7 3 (0,7/3) | Zeros is the x value (that makes the bracket equal 0) in the NOT eliminated term/s in the NUMERATOR. Holes are the x values (that make the bracket equal 0) in the term/s that ARE eliminated. Vertical asymptotes are the x values (that makes the bracket equal 0) that are NOT eliminated in the DENOMINATOR. Y-Intercepts can be found when you substitute 0 for the x values in STANDARD form.

19: How to multiply and divide rational expressions? | 1. Factor both the denominators and numerators. 2. Join numerators from the two rational expressions, and the denominators, together through multiplication 3. Eliminate the common terms from the numerator and denominator 4. Factor out common factors and eliminate to simplify further, then state the restrictions | TO MULTIPLY | To DIVIDE you simply have to flip the second rational expression upside down and multiply like this, just as if you would with a fraction. (switch the numerator with the denominator, get the reciprocal) | 19

20: How do I add and subtract rational expressions? | * Factor if possible 1. Multiply each rational expression (both numerator and denominator) by the term in the other's denominator that is not already in its own denominator. 2. Keep the common denominator (just like with fractions when adding) and add the numerators together. Simplify 3. Factor and simplify as much as possible, then state the restrictions | The same steps are done for SUBTRACTION, except in step 2 you must subtract, not add.

21: 21 | Radicals

22: How to simplify radicals? | 1. Find the largest perfect square 2. Write the numbers multiplying under the radical sign 3. Check for perfect squares and simplify | How to add and subtract radicals? | * Simplify radicals if possible 1. Look for like radicals (radicals with the same number under the radical sign) 2. Add/subtract the coefficients of the like radicals, leaving the number under the sign the same

23: How do I multiply radicals? | * Simplify radicals if possible 1. Multiply the numbers inside radicals, and the coefficients separately 2. Write the product of radicals under the sign, and the product of coefficients next to it | How to use fractions in exponents with rational expressions? | A number raised to a fraction is equivalent to a radical. | There are two methods for solving and a number raised to an exponent: | 23

24: The teacher's method | Insung's method | 1. Write the denominator above the radical sign, and the numerator under the sign. Leaving the negative outside 2. Solve whats in the radical sign, then whats over the radical 3. Solve 1. Think of the base separately first. Put the base in the lowest terms (keep square rooting, or cubed root) 2. Multiply the exponents 3. SImplify

25: 25 | Exponents

26: How to multiply powers? | You can only multiply powers that have the same bases. You simply have to keep the bases the same and add the exponents together. | For example: | How to divide powers? | For example: | To divide powers you have to keep the bases the same, and subtract the exponents.

27: How do to do a power of a power? | To get the power of a power, you have to keep the base the same, and multiply the exponents. | For example: | *When it is raised to a negative number, it becomes 1 over the base to the positive exponent. *When a number or expression is raised to the power of 0, it will equal 1. *When simplifying algebraic expressions involving powers, the answer should always be POSITIVE | 27

28: Exponential Functions

29: How do I apply exponential functions in real life? | It is used to model exponential growth and decay f(x)= ab a - the initial amount (number) b - (1 + growth rate) OR (1 - decay rate) x - the number of growth/decay periods f(x) - the final amount For example: If the population was 15 000 in year 1997 And grows 6% every year, When will the population be doubled? a = 15 000 b=1+0.06 f(x)= 15 000x2 x= ? (year) b=1.06 f(x)= 30 000 | x | 29

30: Trigonometry

31: How to do the reciprocals of trigonometric ratios? | How to graph a sinusoidal function by hand? | You need to use the special points on a sine or cosine curve. Apply the transformations to the 4 points and graph them on the grid. POINT TRANSFORMATION FRMULA: The key points for sine are (0 ,0), (90 ,1), (180 ,0), (270 ,-1), (360 ,0) The key points for cosine are (0 ,1), (90 ,0), (180 ,-1), (270 ,0), (360 ,1) | o o o o o o o o o o | 31

32: How to find the equation of a sinusoidal function? | The equation for a sinusoidal function is or | You need to figure out the following and plug them into the formula Amplitude (a) - distance from the axis of curvature to peak/bottom Period (360/k) - where the pattern starts to repeat (later divide 360 by it) Vertical translation (d) - how many units it was moved up/down Phase shift (c) - how many degrees it was moved to the side | In a parent function they are Amplitude: 1 Period: 360 Vertical translation: 0 Phase shift: 0 Starting point for sine: (0,0), for cosine: (0,1) | Amplitude: 1 Period: 60 Vertical translation: -3 Phase shift: 30 left/right | . | 1 1 | Now we can figure out the equation a - (1) k - 360 / 60 = (6) d - (+/- 30) c - (-3) | f(x)=1sin(6(x+30))-3

33: How to determine the number of triangles possible in an ambiguous case? | If you are given two sides and an angle you have an ambiguous case. The length of the unknown side will determine how many triangles you have. First you will need to calculate the height of your triangle using Pythagorean theorem. | If the unknown side is shorter than the height - a**
**

**34: CHECK YOUR UNDERSTANDING | 1. Are these functions or not? a) b) c) (0,0), (1,1), (2,4), (3,10) d) (x-5) +(y-9) =9 e) 2. Evaluate 3. Get the inverse function of a) b) using different methods.**

35: a) Write an equation for a square root function that as been stretched vertically by 4, moved 3 units to the right, 10 units down, and compressed vertically by the factor of 2. b) What transformations have been applied to this quadratic function? Create a table of values for the two equations in #4. Prove that these equations are equivalent. Solve this polynomial problem Factor using different techniques for each a) b) c) Simplify, solve and state the restrictions a) b) | 35 | 4. 5. 6. 7. 8. 9. 10. | [ ]

36: 11. Simplify 12. Solve 13. Solve d) e) 16. Get the trigonometric ratios and then solve to the nearest tenth: a) b) c) 17. Graph this sinusoidal function f(x) 18. Get the equation of this graph

37: 37 | CONGRATS ;) | 19. If a triangle ABC has - angle (A) that is 30 degrees - side(b) of 15cm - side (c) of 6cm - and side (a) 14cm How many triangles are possible?

38: Answers: 1. a) Not function b) Function c) Function d) Function e) Not function 2. f(6)=9 3. a) b) 4. a) b) horizontal stretch by 3, 10 units up 5. 7. 9.a) b) 8. 10. 11. 12. | x

39: 39 | 13. 2 14. 15. 16. a) 19.1 b) 1.0 c) 14.3 17. 18. f(x)=2sin(6(x-15)) 19. 3 triangles

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