MCR3UO - G Billy

MCR3UO - G 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
     • 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 f(x)
     • the a-value will determine whether there is a min or max value
       • a > 0 means there is a min value
       • a < 0 means there is a max value
   • 1.4
     • when identical radicals are identical and 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 fomula
     • the number of solutions to a quadratic equation and he 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 then be found
       • the roots are placed into the factored form equation: y = (x - r) (x - s)
       • you can then expand and complete the square to get the vertex
     • When given 1 root and a point on the function, the equation can still be found
       • Since only 1 root is given, that root should be the vertex. (x, y) subs into (h, k)
       • plug the vertex values into the vertex equation y = a(x - h)² + k
       • then plug the given point into the x and y value and solve for a
   • 1.7
     • Some linear quadratics may intersect with others, the point of intersection can be found
       • Since the y-values will be the same, the two equations will equal each other
       • then simply 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 a certain value 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 simplify the rational expressions, you would state any restrictions each time you do a step
     • 2.2 part 2
       • if a function consist of addition or subtraction between 2 fractions, then the process is the same as 2.1
   • 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
         • if k is negative, there is a reflections along the y-axis
   • 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
       • y = x - 21