Math Grade 11 Exam Prep PDF

UNIT 1 U1L1: Domain and Range State the domain and range for each of the following graphs, using set notation U1L2: Functions and Function Notation ​ Relation -​ Mapping between a domain and a range -​ Can be represented as: list of ordered pairs, mapping, table, graph, equa...

UNIT 1 U1L1: Domain and Range State the domain and range for each of the following graphs, using set notation U1L2: Functions and Function Notation ​ Relation -​ Mapping between a domain and a range -​ Can be represented as: list of ordered pairs, mapping, table, graph, equation ​ Function -​ A relation where each element in the domain maps to a single element in the range -​ Given any x value there is only one y value associated with it ​ Vertical Line Test (VLT) -​ Used to test if a graph represents a function -​ If a vertical line through any portion of the graph touches the graph more than once, the graph does not represent a function ​ Function Notation -​ Functions can be described using function notation. The linear equation y=-3x +6 is a function. In function notation: f(x)= -3x+6 ➔​ X is the input of functions ➔​ f(x) is the output function ( it does not mean f times x) ➔​ F is the name of the function Ex. Let f(x)= -3x+6. Determine the following: a)​ f(0) b)​ f(-4) c)​ f(a-1) d)​ f(2)-f(1) e)​ 3f(5) Ex. Let f(x)= x^2 + 5x -14. Determine value(s) for x such that f(x) = -20 U1L3 Skill Builder: Operation with Polynomials ​ Polynomial - numerical coefficients are real numbers, exponents are non-negative integers  -​ Monomial - one term -​ Binomial - two terms -​ Trinomial - three terms ​ Degree of polynomial is the value of the highest exponent -​ Polynomial of degree 0 is called a constant -​ Polynomial of degree 1 is called a linear expression -​ Polynomial of degree 2 is called a quadratic expression -​ Polynomial of degree 3 is called a cubic expression -​ Polynomial of degree 4 is called a quadratic expression Adding and Subtracting Polynomials ​ To add or subtract polynomials, combine like terms ​ Remember that if you are subtracting a polynomial, you must subtract each term of the polynomial Ex. Simplify (-2x^2 + 5x -3) (x^2 -6x +1) - (-3x^2 -2x -4) Multiplying Polynomials (FOIL) ​ To multiply (or expand) polynomials, use the distributive property - multiply each term inside the bracket by the number/term outside of the brackets -​ When a polynomial is multiplied by another polynomial, this means that every term in the first polynomial is multiplied by every term in the second polynomial ​ After applying the distributive property don’t forget the collect like terms! Ex. (2y-5)(3y^2 +4y -6) U1L4 Skill Builder: Factoring ​ Factoring involves changing a polynomial relation from standard form (i.e. f(x) = ax^2 + bx + c) to factored form (i.e. f(x)= a(x-r)(x-s) ​ Remember! Always look for common factors first! ​ Then, if a quadratic remains, factor whatever is inside the brackets, if possible Factoring by Grouping (should have 4 terms in polynomial) ​ For groups of two terms -​ Factor each group so that has the same common factor remaining -​ Factor this common factor onto one bracket and the remaining terms into another bracket ​ For groups of three terms  -​ Factor a trinomial into two identical brackets which can then be written as ( )^2. -​ This may result in a difference of squares which can then be factored into two large brackets -​ Simplify the two large brackets as much as possible, eliminating any brackets within Ex. a) xy + 6x +5y +30 b) y^2 -4y +4 -16 x^2 U1L5 : Max and Min of Quadratic Functions Different forms of a quadratic function: FORM GIVES US Standard form: f(x) = ax^2 + bx + c Direction of opening, vertical stretch, y-intercept Vertex form: f(x) = a(x-h)^2 + k Direction of opening, vertical stretch, vertex (found by completing the square) Factored form: f(x) = a(x-r)(x-s) Direction of opening, vertical stretch, zeros or (found by factoring) x-intercepts ​ The maximum or minimum (optimal) value of a quadratic function is the y-coordinate of the vertex. There are a variety of strategies to determine the vertex of a quadratic function -​ Method one: Factoring to determine the zeros and use to determine vertex Ex. Method 2: Completing the square and read vertex (h, k) from equation in vertex form  Ex. U1 L6: Simplifying Expression with Integer Exponents Recall: Exponent Laws Rule Numeric Example Algebraic examples Product 2^3 X 2^4 = 2^7 a^m X a^n = a ^m+n Quotient 5^6 / 5^2 = 2^4 a^m / a^n = a ^m-n Power of a power (3^3)^2 = 3^6 (a^m)^n = a^mn Power of a product (2 X 3)^4 = 2^4 X 3^4 (xy)^m = x^m y^m Power of a quotient (⅗)^2= (3^2 / 5^2) (x/y)^m= (x^m / y^m) ​ Zero Exponent Rule: a^0 =1, a cannot = 0 ​ Negative Exponents -​ Any base raised to a negative exponent equals the reciprocal of the base to the positive exponent a^-n= 1/a^n U1L7: Rational Exponents  ​ Rule: , means the n^th root of x Ex. ​ Rules , mean the n^th root of m^th power of x Ex. UNIT 2 U2L1: Operation with Radicals ​ Simplifying Radicals 1.​ Find 2 factors, one of which is a perfect square (highest perfect square possible) 2.​ Rewrite as two radicals 3.​ First Radical must be the perfect square 4.​ Evaluate the perfect square ex. ​ Multiplying Radicals -​ Whole number times whole number, radical times radical -​ Before every radical is the whole number 1. It’s just not necessary to write it -​ After every whole number is the radical is square root of 1. It’s just not necessary to write it -​ Reduce radical Ex. ​ Dividing Radicals -​ Whole number divided by the whole number, radical divided by radical -​ Before every radical is the whole number 1. It’s just not necessary to write it -​ After every whole number is the radical square root of 1. It’s just not necessary to write it -​ Reduce radical Ex. ​ Adding/Subtracting Radicals -​ You can only add or subtract like radicals (think algebra: like terms) -​ Reduce if needed, then collect like radicals Ex. ​ Simplifying the last step of the quadratic formula 1.​ Reduce radical to mixed 2.​ Factor a common factor in the numerator only if you will be able to use it to simplify the denominator Ex.  ​ Multiplying Radical Expressions -​ To multiply radical expressions use the distributive law (FOIL for a binomial multiplied by a binomial) and simplify where possible U2L2: Solving Quadratic Equations ​ Quadratic equation: An equation of the form ax^2 + bx +c = 0 -​ The solution to a quadratic equation is also called the roots of the equation -​ There are three methods to solve a quadratic equation Method 1: Inverse Operations —> Isolation Use this method when there is a single x-term(vertex form) -​ Use inverse operations to isolate x -​ When you take the square root, recall that there should be 2 answers -​ Leave answer in simplified radical form, unless specified otherwise Ex. Solve 2(x - 9)^2 - 19 = 5 Method 2: Factoring -​ Rearrange equation so it's in standard form ax^2 + bx +c =0 -​ Factor, if possible -​ Set each factor to 0 and solve each linear equation  Ex. Method 3: Quadratic formula -​ Rearrange equation so it is in standard form (ax^2 + bx + c) -​ Substitute a,b, c into the quadratic formula -​ Leave answer in simplest radical form, unless specified otherwise U2L3: Zeros of a Quadratic Function How to determine the number of zeros (x-intercepts) ​ Factored form: f(x)= a(x - r)(x - s) -​ The number of zeros will be equivalent to the number of unique factors -​ If there are no zeros, the equation cannot be written in factored form ​ Vertex form: f(x) = a(x-h)^2 + k -​ If a and k have the same sign = no real zeros -​ If a and k have opposite signs = two real zeros -​ If k = 0 = one real zero  ​ Standard form: f(x) = ax^2 + bx + c -​ Calculate the discriminant ( b^2 - 4) -​ If b^2 - 4 is greater than zero = 2 real zeros -​ If b^2 - 4 is equal to zero = 1 real zero -​ If b^2 - 4 is less than zero = no real zeros( two complex zeros) U2L4: Equation of a Quadratic Function ​ A family of Parabolas is a group of parabolas that share common characteristics -​ Vertex form: where a is varied, this results in a family of parabolas with the same vertex and axis of symmetry -​ Factored form: where a is varied, this results in a family of parabolas with the same x-intercepts and axis of symmetry -​ Standard form: where a and b are varied, this results in a family of parabolas with the same y-intercept Ex. Write the equation ( in standard form) of the quadratic function that passes through the point (-1,3), if the roots of the corresponding quadratic equation are U2L5: Linear/ Quadratic systems ​ A linear system involves 2 linear functions with the same independent and dependent variables. The solution of the linear system is the point of intersection (POI) of the 2 lines. Linear systems can be solved graphically or algebraically (substitution or elimination) ​ A linear- quadratic system involves one linear function and one quadratic function. The solution of the system is the point(s) of intersection of the 2 functions. There may be 0,1, or 2 solutions Steps to substitution: 1.​ Isolate y in the linear equation 2.​ Sub into the quadratic equation 3.​ Solve the quadratic (factor or quadratic formula) 4.​ Sub. each x-value back into the line to get y Unit 3 U3L1: Simplifying Rational Functions A rational function is the ratio of two polynomial functions. A rational function can be expressed as R(x)= p(x) / q(x) ​ To determine the domain of a rational function, consider the value(s) of x that make the polynomial in the denominator, q(x)=0(i.e. The zeros of the denominator). The domain will exclude these values. These values are also called the restrictions of the corresponding rational expressions -​ We can simplify rational functions and rational expressions in a similar manner -​ Factor both the numerator and the denominator (using all of your factoring strategies) -​ Divide both numerator and denominator by the GCF (“Cancel out” all common factors) -​ When asked for the DOMAIN, you must determine the zeros of the denominator Ex. Simplify f(x) and state the domain ​ Equivalence -​ Two functions are considered equivalent if they have the same domain and yield the same values(output) for all numbers in their domain (input) -​ To show equivalence, you must show that they both simplify to the same expression with the same domain -​ To show non-equivalence, you can choose an input(i.e. Substitute a number for “x”) and show that each function yields a different output. This does not work to show equivalence since some functions intersect Ex. Determine if it is equivalent U3L2: Multiplying and Dividing Rational expressions ​ Recall: The procedure for multiplying numerical fractions -​ Check all the numerators and all the denominators for common factors -​ Divide out all common factors (“cancel out” common factors) -​ Multiply numerator by numerator and denominator by denominator ​ We can multiply rational expressions in a similar manner -​ Factor the numerator and denominator of both rational expressions -​ Divide out any factors common to the numerator and denominator (“Cancel out” all common factors) -​ Multiply numerator and denominator by denominator ➔​ You do not need to expand your final expression (Leave final answers in factored form) Ex. Multiply. State any restriction on the variables  ​ Recall: The procedure for dividing numerical fractions -​ Take the reciprocal of divisor ( the 2nd fraction, the one you are dividing by) and change the division sign to a multiplying sign -​ Proceed with the same steps as multiplying ​ We can divide rational expressions in a similar manner -​ Take the reciprocal of the divisor (the 2nd rational expression) and change the division sign to a multiplication sign -​ Factor the numerator and the denominator of both rational expressions -​ Divide out any factors common to the numerator and denominator (“cancel out” all common factors) -​ Multiply numerator by numerator and denominator by denominator ➔​ You do NOT need to expand your final expression. Leave final answers in factored form Ex. U3L3: Simplifying Algebraic Expressions with Exponents Rule Numeric Example Algebraic examples Product 2^3 X 2^4 = 2^7 a^m X a^n = a ^m+n Quotient 5^6 / 5^2 = 2^4 a^m / a^n = a ^m-n Power of a power (3^3)^2 = 3^6 (a^m)^n = a^mn  Power of a product (2 X 3)^4 = 2^4 X 3^4 (xy)^m = x^m y^m Power of a quotient (⅗)^2= (3^2 / 5^2) (x/y)^m= (x^m / y^m) Zero exponent 4^0 = 1 a^0 = 1, a does not = 0 Negative exponents 6^-2 = 1/ 6^2 a^-n = 1/ a^n Rational exponents 8^⅔ = (3√8)^2 X m/n = (n√x)^m ​ When simplifying expressions involving exponents follow the laws and rules for exponents and the order of operations ​ Rewrite any decimal exponents as fractions ​ Rewrite numbers as powers with the same bases, if possible ​ Express all final answers using positive exponents ​ Express all answers in rational form (no decimals) Ex. U3L4: Solving Equations with Exponents Recall: Solving any equation means find the value of the variable that makes the equation true. When solving equations involving equations, pay attention to the location of your variable in the equation. ​ Variable already isolated -​ Apply correct order of operations (exponents before multiplying) and evaluate ​ Variable is being multiplied by a power -​ Solve using inverse operations ​ Variable is the base of a power -​ Use inverse operations to isolate the power -​ The exponent in the power becomes the type of root needed to solve for the base Ex. ​ Variable is the exponent 1.​ Strategy 1 — Guess and check -​ Since we don’t know how to “undo” the raising of a base to an unknown variable, we can use a guess and check Ex. 2.​ Strategy 2 - Change of Base -​ Consider the equation a^x = a^y. Since the bases are equal, it follows that the exponents must be the same as well -​ Steps to follow: ➔​ Rewrite all powers with a common base  ➔​ Simplify to get a single power on each side of the equation ➔​ Create a new equations with the exponents ➔​ Solve the new equation to get the solution(s) of the original equation Ex. Unit 4 U4L1A: Solving Triangles - SOH CAH TOA “Solving a Triangle” means to determine the lengths of all the sides and the measure of all the angles. Unless otherwise stated, round all lengths to one decimal place and angles to the nearest degree ​ Solving right triangles -​ If you are solving a right triangle, use the primary trig ratios and the Pythagorean theorem (Using sine or cosine law will cost you technical marks)  ​ Note: Angle of elevation: Angle of depression: U4L2: Trig Ratio and Special Triangles Recall: Primary Trigonometric Ratios (SOH,CAH,TOA) New: Reciprocal Trigonometric Ratios Ex. Determine the 6 trigonometric ratios for Ө ​ Special Triangles -​ Special Triangles are used to determine the exact ratio for certain special angles (no calculators) ➔​ 45-45-90 triangles ➔​ 30-60-90 triangles  ​ Rationalizing the denominator is a process that is used to change a rational expression so it does not have a radical in the denominator. It uses the identity property of 1 (a number multiplied by 1 retains its value) Ex. U4L3: Trig Ratios for Angles Between O∘ and 360∘ ​ An angle has 3 parts: Initial arm, vertex, terminal arm -​ For an angle to be in standard positive x-axis -​ Vertex must be at the origin -​ Angle is measured from initial arm to terminal arm ​ The principal angle (Ө) is the counterclockwise between the initial arm and the terminal arm of an angle in standard position. Its value is between 0 and 360 ​ The related acute angle (β) is the acute angle between the terminal arm of an angle in standard position and the x-axis when the terminal lies in quadrants 2,3, or 4 ​ CAST: -​ Positive angles are formed by a counterclockwise rotation of the terminal arm -​ Negative angles are formed by a clockwise rotation of the terminal arm  Ex. ​ Co-terminal angles have the same initial arm and the same terminal arm but have different angle measurents Ex. Determine 3 angles that are co-terminal to 45∘ U4L4: Solving Trig Equations Recall: SinӨ = o/h, cosӨ = a/h, tanӨ=o/a For point P(x,y) which lies on the terminal arm of an angle in standard position SinӨ = y/r, cosӨ = x/r, tanӨ=y/x Where r^2 = x^2 + y^2 Ex. The point Q(6,-3) lies on the terminal arm of an angle in standard position ​ When solving for an unkown angle, you must consider all values that would make the equation true! 1.​ Consider the ratio (sine, cosine, tangent, cosecant, secant, cotangent) and its sign (+/-) to determine the quadrants where your angles will terminate 2.​ Draw a sketch of the 2 angles 3.​ Solve for β, either with your calculator or using special triangles 4.​ Use β to determine θ1 and θ2 Ex. U4L1B: Solving Triangles using Sine and Cosine Laws ​ Solving oblique triangles -​ An oblique triangle is a non-right triangle that is either acute (all angles 90). To solve an oblique triangle, youse must use either the sine law or the cosine law. When labelling triangles, angles are uppercase letters, sides are lowercase. Sides and angles opposite each other share the same letter ​ Sine law is used to determine: -​ The length of the side of a triangle if you are given any two angles and 1 side -​ An angle if you are given two sides and an angle opposite one of these two sides -​ The 2 forms of sine law: Ex. ​ Cosine Law is used to determine: -​ The length of the side of a triangle if you are given two sides and the contained angle -​ Any angle if you are given three sides -​ The three forms of coisne law:  ​ When solving a triangle given 3 sides always start by using cosine law to find the angle opposite to the largest side -​ Use cosine to determine the largest angle -​ Then use sine law to determine either of the two remaining angles -​ Subtract the two known angles from 180 to determine the third U4L5: The Ambigous Case of the Sine Law The primary Trig ratios are useful for solving right triangles, but NOT oblique triangles (Triangles that are either acute or obtuse) Sine law can be used if you know two angles and any sides (AAS or ASA) , or two sides and one angle opposite a given side (SSA) ​ SSA: GIven angle is obtuse. There may be 1 triangle or no triangle possible to determine the number of possible triangles, you must recall an important property of triangles: -​ The smallest side is across from the smallest angle and the largest side is across from the largest angle 1. If a > b, 1 triangle 2. If a ≤ b, no triangle Ex. ​ SSA: Given angle is acute, may be 2 triangles, 1, or none 1.​ If h> a, no triangle 2.​ If h=a, 1 right triangle 3.​ If h (x,ay) a If a (x/k,y) of 1/k If k (x+d, y) c Vertical translation shift (x,y) —--> (x, y+c) U5L5: Graphing Functions with Transformations - Exponentials An exponential function with base B that has been transformed has the form: g(x) = aB^k(x-d) +c Ex. U5L6: Graphing Functions with Transformations - Sinusoidals A sinusoidal function that has been transformed has the form: g(Ө) = asin[k(Ө-d)] + c or g(Ө) = acos[k(Ө-d)] + c ​ The amplitude is l a l ​ The period is 360/k ​ The # of cycles (numbers of times a graphs repeats within the domain of the base curve) is k ​ The equation of the axis is y=c ​ The vertical displacement is c ​ The range is affected by the amplitude and the vertical displacement Ex. U5L7: Graphing Sinusoidal Functions using Characteristics ​ The amplitude is l a l ​ The period is 360/k ​ The # of cycles (numbers of times a graphs repeats within the domain of the base curve) is k  ​ The phase shift is d (remember to factor out k if needed) ​ The equation of the axis is y=c ​ The vertical displacement is c ​ The range is affected by the amplitude and the vertical displacement U5L8: Representing Functions with Equations Given the graph of a function, you can determine the transformations that have happened to the parent function to determine its equation 1.​ Identify the parent function by observing the shape of the graph ​ Each parent function has a distinct shape ​ For exponential functions, this includes determining the base B if not given in the question 2.​ Identify any translations (shifts) that have occurred ​ It’s often easiest to look at the vertex or the asymptotes, since those are not affected by dilations ​ Write the equation of the transformed function with the appropriate shifts 3.​ Identify any reflections that have occurred. Represent those in the equation 4.​ Finally, chose a point on the graph (not the vertex) and substitute its coordinates (x,y) into your equations to solve algebraically for either a or k Ex. The graph has undergone a transformation in the form g(x) = f[k(x-d)]+c. Determine the equation of the transformed function U5L9: Representing Functions with equations (sinusoidal) For graphs of sinusoidal functions, we are often interested in obtaining the simplest equation. This means: ​ Avoid phase shifts(horizontal translations), if possible -​ Sometimes you can use a reflection in the x-axis instead of a phase shift -​ If possible, choose a base curve (sine or cosine) that avoids a phase shift ​ Avoid using reflections in the y-axis Parameter Property How to determine/ calculate a Amplitude max- min / 2 a