"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

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Transformations:

Lesson 1:

• Be able to recognize interval notation: ( #, *) means that the endpoints are not included [#,*] means that the endpoints are included. An endpoint of infinity (positive or negative) is never included.

• Know the general forms, domain, range, and intercepts of the six foundation graphs (page 3 and 4). All of the graphs you will work with are related to these six graphs.

• For the equation y = f(x) if y is replaced with y-k that means the graph moves up by k units. If y is replaced with y+k that means the graph moves down by k units.

• For the equation y = f(x) if x is replaced with (x-h) that means the graph shifts right by h units. If x is replaced with (x+h) that means the graph shifts left by h units.

Lesson 2:

• Replacements

  • know that you may be given information about the direct replacement of y or x (this relates to their replacement in the equation y = f(x))​

    • y --> ay means a vertical stretch by a factor of (1/a)​

    • x --> bx means a horizontal stretch by a factor of (1/b)

• Function notation - use replacement information to find the equation of an "image" or transformed graph

  • start with y = f(x), directly replace y and x with the replacement information (see above), rearrange the equation to isolate y on the LHS. This is the equation of your ​image or transformed graph.

• Mapping notation - use this to find points on the translated graph if you are given points from the original graph. BE CAREFUL the mapping notation is NOT the same as the replacement.

  • (x,y) --> (x, (1/a)y) means a vertical stretch by a factor of (1/a)​

  • (x,y) --> ((1/b)x,y) means a horizontal stretch by a factor of (1/b)

• Invariant points - know which points will always stay the same in a stretch (this is due to a multiplication of the stretch factor by zero)

  • vertical stretch --> x-intercepts do not change​

  • horizontal stretch --> y-intercepts do not change

Lesson 3:

• Vertical Reflections

  • replacement: y --> -y

  • ​function notation: y = -f(x)
  • mapping notation: (x,y) --> (x,-y)
  • invariant points: any x-intercept
• Horizontal Reflections
  • replacement: x --> -x​
  • function notation: y = f(-x)
  • mapping notation: (x,y) --> (-x,y)
  • invariant points: any y-intercept

Lesson 4:

• Combining Functions - Order of applying transformations:

  • FIRST apply scale (stretch and compress) either vertical or horizontal​

  • SECOND apply reflections (deal with the negatives) either vertical or horizontal

  • THIRD apply translations (deal with addition and subtraction) either vertical or horizontal

  • FOR EXAMPLE: y - 7 = -2f(3x-6) --> y - 7 = -2f(3(x - 6)) --> y = -2f(3(x - 6)) + 7

    • first scale​

      • horizontal stretch by 1/3​

      • vertical stretch by 2

    • second refelct

      • vertical reflection​

    • third translate

      • horizontal shift to the right by 6​

      • vertical shift up by 7

• Mapping notation

  • order matters, make sure the scale is applied first, the reflection is applied to the scale, and the shift is applied to the reflected scale.​

    • FOR EXAMPLE (from above)

      • first scale​

        • (x,y) --> ((1/3)x, 2y)​

      • second reflect

        • ((1/3)x, 2y) --> ((1/3)x, -2y)​

      • third translate

        • ((1/3)x + 6, -2y + 7)​

Lesson 5:

• To find the inverse of a function, switch the "x" and the "y" in the function equation (where y = f(x)) and solve for your new "y" we call this new "y" the inverse and the equation for the new "y" is the inverse function (add a "-1" exponent to f(x) to show that this is the inverse function)

• the invariant points for inverse functions occur on the line y = x

  • For example, if you have a point on y = x, say (3,3) and you replace the x values with the y values and vice versa, then you are just replacing a 3 with another 3 so nothing changes.​

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Combinations:

Lesson 1:

• Adding and subtracting functions: the "input" (x) is the same for the parent functions and the new combined function. The "output" (y) is the algebraic result of combining the output of the parent functions

  • For example: if f(x) = x + 2 and g(x) = 2x +3 and h(x) = f(x) + g(x) then at the value x = 4, f(x) = 6, g(x) = 11, and h(x) = 4 + 11 = 15.​

• The domain of a combined function by addition or subtraction is equal to the overlap of the parent functions

• The range of a combined function by addition or subtraction is found by inspecting the graph of the new function

• Problems can be solved three ways:

  • make a table of values​

  • inspect the graph (the y-coordinates are the "output")

  • use your calculator (y1 = f(x), y2 = g(x), y3 = y1 + y2) --> use y=, VARS, Y-VARS, 1:Function, then select Y1 or Y2 as input values for Y3.

Lesson 2:

• Problems can be solved in 3 ways (see lesson 1)

• The output (y-values) are what you are manipulating, the x-values do not change

• For division, use your knowledge of rational functions (horizontal asymptotes, vertical asymptotes, points of discontinuity) to find restrictions on the domain and range

• When in doubt for the domain and range, graph the function and investigate!

Lesson 3:

• A table is very, very useful for composite functions. Have one column for the "x" values, one column for your "input" function (the one in the inner parenthesis), and one column for your "output" function (the one that takes the values from the second column and applies them to the new function)

• make sure you know both ways of writing composite functions: g(f(x)) = g o f(x)

• There are two ways to solve composite functions:

  • write out the "output" function (the one that is outer-most in parenthesis form) leaving a space wherever there is a variable --> fill up each space with the "input" function (the inner-most in parenthesis form). Simplify where necessary and substitute if you are given a value for your variable.​

  • If you have a value for your variable, substitute and solve for the "input" function (the inner-most in parenthesis form) --> take your new number and substitute it for the variable in the "output" function (the outer-most in parenthesis form)

• For restrictions, look at the restrictions of the inner-most function, and then apply any "special" restrictions from the outermost function (for example, if the outer-most function involves a square root, you need to ensure the radicand is greater than or equal to zero).

• Order matters f(g(x)) might not give you the same result as g(f(x))

• If f(g(x)) = g(f(x)) then the two functions are inverse functions.

Lesson 4:

• The easiest way to sketch the graph of a composite function (for example, f(g(x)) is to make a table of values with the input (x), inner-function (g(x)) and outer-function f(g(x)), then plot the points (x,f(g(x)))

• The range of a composite function is best found by observing the graph of the composite function

• An explicit equation is an equation with the variables separated  on either side of the equal sign

  • for example, y = 5x +1 is explicit, 3 = 4y +6x is not explicit

• Whenever you are dealing with a rational function, make sure that you factor everything and write it in the simplest form, that way you can observe restrictions (horizontal asymptotes, vertical asymptotes, points of discontinuity)