MCR3UO - Billy

MCR3UO - Billy

1. Unit 1 Functions

   • 1.1
     • A relation is a function if for each x-value has only one y-value
     • an x-value with more than one y-value is not a function
     • domain and range of a function can be found from the restrictions
       • restrictions on the domain is found from a fraction function because you cannot simply sub an x value and divide by 0
       • range is found if a y-value cannot exceed a certain point on the graph
   • 1.2
     • The symbol f(x) represents the dependent variable
       • f indicates that it is a function and the outcome relies on what x equals
         • it does not always need to be represented by f
       • eg. f(x) = 3x
         • if x = 1, then y = 3
         • if x = 3, then y = 9
   • 1.3
     • the a-value will determine whether there is a min or a max value
       • a > 0 means there is a min value
       • a < 0 means there is a max value
     • finding the max or min of a quadratic function can be found by completing the square or partial factoring
     • completing the square f(x) = a(x - h)² + k. The vertex is (h, k)
     • Partial Factoring
       • is simply when you find the mid point of 2 x values
   • 1.4
     • when identical radicals are identical and are multiplying each other, they can be set as a full term
       • √5 x √5 = 5
     • Radicals follow the like term rule
       • 4√2 - 2√2 = 2√2
       • 5√2 + 4√2 = 9√2
     • Values inside radicals can be broken apart into values that have the product of it
       • √20 is same as √5 x √4
   • 1.5
     • Quadratic equations can be solved by
       • Completing the Square
       • Factoring
       • using the quadratic formula
     • the number of solutions to a quadratic equations and the number of roots of the related quadratic function can be determined using the discriminant
       • b² - 4ac > 0 then there are 2 roots
       • b² - 4ac < 0 then there are no roots
       • b² - 4ac = 0 then there is only 1 root
   • 1.6
     • When the 2 roots are found and a point is given, the quadratic equation can be found
       • the roots are placed into the factored form equation: y = a(x - r) (x - s)
       • you can then sub in the 2 roots into r and s and isolate for a by subbing in the given point
     • When given 1 root and a point on a function, the equation can still be found
       • the process is the same but since there is only 1 root, the vertex is given so you would sub in the equation y = a(x - h)² + k and isolate for a
   • 1.7
     • Some linear quadratics may intersect with others, the point of intersections can be found
       • since the y-values will be the same, the two equations will equal each other
       • then simplify each equation by setting the values to one side
       • using the discriminant, the number of intersections will be found
         • the same rules apply as if the discriminant was used to find the roots

2. Unit 2 Transformation

   • 2.1
     • complex functions can be the same as simpler functions
       • You would first state any restrictions at the beginning
         • Since in a fraction, if the denominator is 0, then the value is nothing. So restrictions must be assigned to certain values so that the denominator does not equal to 0
           • 1 —— 2-x
             • x ≠ 2
       • Then simplify the function fully
       • Then state any restrictions
   • 2.2
     • 2.2 part 1
       • if a function consist of division or multiplication between 2 fractions, then the process is different
       • You would first state any restrictions at the beginning
       • Simplify the function
         • As you are simplifying the rational expressions, you have to state any restrictions each time you do a step
     • 2.2 part 2
       • if a function consists of addition or subtraction between 2 fractions, then the process is the same as 2.1, but you would have to do common denominator to do any simplifying
   • Combinations of Transformations
     • y = af (k (x - d) ) + c
       • a-value
         • The a affects the function by stretching or compressing vertically by a factor of a
         • if the a-value is negative, there is a reflection along the x-axis
       • d-value
         • the d affects the function translation horizontally
       • c-value
         • the c affects the function translation vertically
       • k-value
         • the k affects the function by stretching or compressing horizontally by a factor of 1/k
   • inverse functions
     • To put it simply, the domain and range is flipped
       • This means the x-values and y-values are switched
     • when forming the equation, you would switch the x and y variable and algebraically solve for y
       • y = 4x + 2
       • x = 4y + 2
         • x and y are switched
       • x - 2 —— = y 4
         • y is isolated
       • x - 2 —— = f-(x) 4
         • the inverse function would be then replaced with f-(x)
   • Transformation equation
     • original function co-ordinates
     • (x/k) +d
       • x from original function and k, d from equation
     • ay + c
       • y from original function and a, c from equation

3. Unit 3

   • 3.1 and 3.2 and 3.4
     • exponential functions are modeled by y = ab^2
       • The function will increase if
         • if a is positive and b > 1
         • if 0 < b < 1 and a is negative
       • The function will decrease if
         • if 0 < b < 1 and a is positive
         • if a is negative and b > 1
       • b can determine the increasing and decreasing of the graph
   • 3.3
     • • square roots means n = 2
         • so if m = 1, you can also say a^1/2
       • This can be very helpful in simplifying and expanding
         • eg. if n = 2 and m = 4, then the rational expression will be a^2
   • 3.5
     • general equation for an exponential function is
       • • if d > 0, translate right if d < 0, translate left
         • if c > 0, translate up if c < 0, translate down
         • if a > 1, stretch vertically by a factor of a if 0 < a < 1, compress vertically by a factor of a if a < 0, reflect along the x-axis and stretch or compress
         • if k > 1, compress horizontally by a factor of 1/k if 0 < k < 1, stretch horizontally by a factor of 1/k if k < 0, reflect along the y-axis and stretch or compress
         • be aware that some exponential functions can be written in different bases but mean the same as others
           • y = 2^4x is same as y = 16^x
         • a horizontal asymptote is a line where it never goes to a natural number and always a fraction
           • Horizontal asymptote y = c
   • Doubling, half life, and other problems
     • you can use the exponential functions to determine half life or doubling in a period of time
       • eg. there is 1 bacteria. The population of bacteria doubles every 3 hours
         • variable a would be the starting population 1
         • m would be the the ratio, in this case it is 2
         • since it is every 3 hours, we would not set n to 3. We would write the exponent of t/n and sub n there
       • y = a(m)^n would be the equation
         • sometimes n is replaced with t/n because if the time frame each action is done

4. Unit 4 Trigonometry

   • 4.1
     • Unit Circle
       • The unit circle is a circle with a centre at the origin and a radius of 1 unit
       • if you consider the triangle within the unit circle, you can state the primary trigonometric ratios easily
         • Opposite Sinθ = ———— Hypotenuse
         • Adjacent Cosθ = ———— Hypotenues
         • Opposite Tanθ = ———— Adjacent
     • Special triangles
       • Exact Trigonometric​ ratios for special angles can be determined using special triangles
   • 4.2
     • Alternatively, you would be given angles that are at different quadrants, and each quadrants have negative and positive trigonometric ratios
       • if you were given an angle higher than 90°
     • but related angles can refer back to a reference angle
       • eg. 120° can be referenced to 60°
     • Once you have the reference trigonometric ratios, you can determine which are negative using the CAST method
       • Each Quadrant represent which trigonometric ratio will be positive and the rest will be negative, however if the trigonometric is at quadrant 1, as you can see quadrant 1 is labelled as A (All) so all trigonometric ratios will be positive
   • 4.3
     • Reciprocal Trigonometric Ratios are defined as follows
   • 4.4 and 4.5
     • An important note to keep in mind are the ambiguous cases of the sine law
       • Acute Trangles
         • if a ≥ b, one triangle exists
         • if a < b and a < bsinA, no Triangle exists
         • if a < b and a = bsinA, one Triangle exists
         • if a < b and a > bsinA, 2 triangles exists
       • Obtuse Triangles
         • if a ≤ b, no triangles exists
         • if a > b, one triangle exists
     • When solving 2d and 3d questions, often refer back to sine law, cosine law and special triangles
   • 4.5
     • Trigonometric Identities require a high algebraic level of skill. As long as you know the following trig identities, more complicated trig identities can be solved.

5. Unit 5 Trig Functions

   • 5.1
     • a period is the horizontal distance of one cycle on a graph
     • Periodic function has a pattern of y-value that repeats at regular intervals
     • axus if the curve is a horizontal line that is hallf-way between the max and min
     • Phas shift is the horizontal shift of a graph from its original position
     • finding the period would be getting 2 x co-ordinate of the max or min of the cycle. you would then subtract each other finding the period
     • the formula for finding amplitude
       • max - min y = ————— 2
     • equation of the axis
       • max + min y = ————— 2
   • 5.2
     • Graphing a sine function are based on these key co-ordinates because we can use the trigonometric ratios of special triangles. The same goes for cosine functions
     • Sine Graph | | | | | | | | |
     • Cosine Graph | | | | | | | | |
   • 5.3 and 5.4
     • y = a Sin[k (x-d)] + c
     • a - amplitude k - period d - horizontal transition c - verticle transition
     • y = a Cos[k (x - d)] +c
     • 360 period = ——— |K|

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