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Unit One - AP Calculus, Graphing Circles Defined by Parameters, T may also…

- - - - Horozontal and Vetrical Lines
        
        Vertical Functions
        
        Horizontal Functions
      - Perpendicular and Paralell Lines
        
        Perpendicular Functions
        
        Lines that are perpendicular are defined by the equation (M1)(M2) = -1 EX. y=mx+b --> y1 = -1x+3 y2= 1x +5 are perpendicular because m1xm2 = -1 (1x-1=-1)
        
        Parallel Functions
        
        Lines that are parallel to each other have the same slope: M1=M2 EX. y=mx+b --> y1 = 2x+3 y2= 2x +5 are parallel because m1=m2 2= 2
        
        Both perpendicular and parallel lines rely on the slope value (m in slope intercept equations)
      - Finding Increments of a Line: Should point (x1, y1) on a line of f(x) shifts vertically or horizontally to become g(x), to find the vertical and horizontal movement of the point, (now referred to as (x2, y2), you would use Δy2-Δy1 and Δx2-Δx1
        
        Understanding how to identify vertical and horizontal movement of a line can help to further explain Slope. To find the slope, remember "rise over run". When finding the slope of a linear equation, often, two points on the line are picked to represent (x1, y1) and (x2, y2). The slope is found using a fraction comprised of the "finding Increments of a line" equations: (Δy2-Δy1)/(Δx2-Δx1)
  - - - Point slope equations can be used to help find the coordinates of a specific point (P(X1, Y1)) which a line intersects. Often questions give incomplete information about P(X1, Y1), to find the coordinates of the line, find slope (m) and use equation: (y-y1) = m(x-x1).
    - - C = Ax + By where A, B ≠ 0 and C represents the function output. Given x and y coordinates are substituted for A, and B respectively. By values are replaced by the b value to equate to Ax + b = C < represented as y
        Ex. a = π b= -5
        Ax + By = C --> Y = πx -5
- - - - "Mapping"
        
        Mapping is especially helpful for when you are trying to identify, or induce a shift in a line between f(x) and g(x). Mapping clearly displays the original points from F(x), and the same set of points located on g(x). This way, to find the change in x and y, and eventually understand the changes the line experienced, you can use x2-x1 and y2-y1. This will tell you how much the points on f(x) moved.
        
        This is similar to how we find increments of a line, as we are finding delta y and delta x from points 1 and 2
        
        Depending on what you are using Mapping for, it can be very similar to a table of . If you are not trying to find a change in a line, you can use it simply for connecting the dots between your x1, x2, x3, x4, values with their corresponding y values. Ex.
      - Equation Form
        
        Ex. y=mx+b is a way of showcasing the slope and y intercept of a particular line. This is one of the easier ways for me to interpret lines. Often in a problem not involving a visual graph, this format will be given. It can help to change the equation of y=mx+b to another one of the equation forms if you are trying to find an unknown variable
        
        The other equations to represent a line are present here: my preferred equation is y=mx+b, then point slope as they both are simpler ways for me to visualize the functions in a graph form (I am super visual). It is easier for me to picture a point on a graph, or the general format of the line from knowing the slope and y intercept.
      - Graphing
        
        Often to graph, you need to find the points first. To do this, I would normally use my graphing calculator and input the function into Y=, or if I was to do it manually, I would use a table of values, then plot the points. If you are given a graph to interpret, you can use y=mx+b to find the simplest form of the function.
        
        To find M (slope)
        
        To find B, the y intercept
        
        Ex. PIC 4
      - Table of Values
        
        A Table of Values is a great way to keep track of your points (relations), and to understand the pattern of the line, especially if your goal is to graph the function. This way, you can use a function machine to obtain the x and y values, then once the desired x values are calculated, you can plot. Also, a table of values can be found on your graphing calculator after entering a function into Y= Calculator Instructions
        
        EX. PIC 3
      - "Function Machine"
        
        F(x=a) = b a function machine can be used to inverse an equation in a simple way. In my opinion it is not ideal, I would rather use any one of the other options, as I easily lose track of individual numbers if I can't see them al at once.
      - Interval Notations: Closed = filled in, Open is not
        
        Interval notation defines the permissible values on a line when the function is graphed
    - - does the function pass the odd equation f(-x)=f(x)
        or the even equation f(-x)=-f(x)
        
        Ex. PIC 5
        
        Piece Wise Functions: You are often given multiple functions for small segments of a line in it's overall domain. (A bit similar to line increments)
        
        Find each part of the line and plot it, then figure out if the line is connected, or if there are non permissible values. Use interval notation.
        
        Ex. PIC 7
        
        Composition of Functions
        
        Sometimes, if two functions (fx, and fg) can intersect if they share both a domain and range. In this case, to solve, we can substitute one equation into another. This can be represented by writing fog(x) or f(g(x)) (where x is the given value of x)
        Ex. Pic 13
      - Vertical Line Test
        
        Use this for a graph, if there are 2 y values for1 x value, it is not a function
      - Odd and Even Functions
        
        Odd
        
        Graphically: be careful with graphically, you cannot just judge if it is odd or even by looking at it, I made this mistake on my unit 1 test, you have to test it both ways in order to be 100% sure in my experience. If the line does not have an equal negative value on the other side of the y axis, it is not even. Ie. F(-x) = F(x)
        
        Pic 5 again
        
        Even
        
        Graphically
        
        If the function is even around the y axis, it is even ie. for every f(x) there is the same point, just negative f(-x). It will often look like a parabola
        
        Pic 6 New Notations: Iff and "for every there is a "
        
        Algebraically
        
        first, take your original function, then make all the x values negative. Then distribute the exponents that act on the x values. If the x values are all positive after distribution, it is an even function, if they are negative, it is an odd function. Should there be a mix of both, if its neither even or odd
        
        Ex. This function is even because when set to the positive version of itself, it is equal after distribution
        
        You can also look at the leading coefficient to see if the exponent is even or odd, this can give you an idea Ex. Y = x^2 + 4 this is even because 2 is even
        
        F(-x) = -f(x) This would mean that for each (x,y) point on the graph, there is an equal negative point
- - - - Domain and Range
        
        D= XER (where X can have all real number values)
        
        R = Y>0 (where the vertical asymtote is 0
      - Horizontal and Vertical Intercept
        
        Vertical and horizontal intercepts can be found by substituting X or Y as 0 and solving for the other variable. Sometimes there are vertical or horizontal asymptotes which prevent intercepts from occurring. in this case you write DNE (does not exist)
      - Asymtote
        
        An asymptote outlines where a non permissible value is on a graph for a certain function. Both a horizontal and vertical asymptote can often be present in specifically exponential functions, as well as rational functions. Asymptotes can be found algebraically, or by looking at a graph (though algebraically is usually more accurate in my opinion). Instead of basing your answer off of a visual check of the graph, maybe use it to double check your algebraic answer.
        
        Good Website for a Detailed Explanation on Rational Asymtotes
        
        Ex. Pic 14
- - - - Parameters: When the x and y values of a relation are dependant on (are functions of) a third variable (t), this function is considered a parametric function as the variable "t" is called a "parameter" . When parametric functions are graphed, one must understand the value of t, to find the values of each x and y (relations) points.
        
        EX. (x,y) = (2,4) when f(x) = x-t and f(y)=y+t. To graph, the value "t" must be found. "t" may also have it's own function. If the given value of the parameter "t" in this question is t=2) then x = x-2 and y = y+2
        
        Without the basic rules of graphing, it is difficult to transition from linear functions, parabolic functions etc. that rely solely on x, y, and domain and range, to introducing the effects of a parameter on such functions. To graph parametric functions, a calculator set to "parametric mode" can be utilized, or a table of values which incorporates a "t" column can also be used. Such a table can be useful when "t" is a higher value as it can make the substitution of the t value into the functions of each relation (x and y) more straight forward for visual learners.
        
        Graphing
        
        To graph an exponential function, the properties of an exponential function are taken into consideration.
        Ex. Y=4^(x+ 3)
        In this instance, we can figure out that there will be a
        shift of 5 up, (+5), and a horizontal shift of 3 to the left (x+3).
        In addition to this, our base (b), "4" shows that the function will be "exponential growth", rather than decay because... 0<b<1 = decay and b>1 is growth! Pic16
        the vertical asymptote is at +5, and the horizontal asymptote appears to be at 0, as the line gets closer and closer, but does not touch
        
        SET EQUATIONS FOR REAL LIFE:
        
        3 more items...
        
        EX. (pic2)
        
        GRAPHING PARAMETRIC FUNCTIONS
        
        Graphing Line Segments
        
        Parametrizing a line segment
        
        "Finding a parametrization for the line segment with given endpoints"
        
        Ex. end points = (4, 3) and (5, 8)
        Step1. Think of these two points as X1 and X2 in a table of values.
        ie. T/X/Y
        0/4/3
        1/5/8
        So, when T is 0 you get your first given point, when T is 1, you get your second. From here, you can use your ΔY and ΔX values to find two functions that model X and Y
        ie. ΔX = 5-4 = 1
        ΔY = 8-3= 5
        so you can say...
        when t is 0, x is 4 +a(t) when t is 0, y is 3+5(t)
        functions are
        x = 4+1(t) and
        y = 3+5(t)
        Test: t=0
        X -> 4=4+0=4 YES
        Y-> 3=3+5(0)=3 YES
        Test t=1
        X -> 5=4+1=5 YES
        Y-> 8=3+5(1)=8 YES
        
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        Ex. To graph a line segment, you would use similar steps as graphing a piecewise function. Use a parametric table of values (T, X, Y) and plot your points within the given domain
        
        Definitions:
        
        Cartesian equation = the equation of a line that eliminates the parameter value "t"
        
        Parametrized curve = curve that is expressed as being dependant on value "t"
- - - - These log rules explain often why we have to "take the log" of a function to solve for an exponent. They also illustrate how logs are the inverse of exponents
        
        Exponential functions and Logarithmic functions are the inverse of each other.
    - - Natural Logarithm --> Log base e --> represented verbally as "lon" --> calculator: "ln"
      - Common Logarithm has base 10, on a calculator, you can assume the log is base 10
  - - - Inverses: One to One functions are Inversible. Ex. (1,2) --> (2,1) and the line will stay the same. The function does not have 2 y values for one x value.
    - - the horizontal line test works when the function is graphed, this is a visual test to see if the function has two y values for one x value
- - - - Finding Trigonometric Ratios: Use SOH CAH TOA to figure out the ratios: Sin = opp/hyp Cos = adj/hyp Tan = opp/adj
        
        To find the trigonometric ratios,
        on the unit circle, revert
        adj = initial arm
        opp = arc length line
        hyp = radius
        
        Ex.
      - Remember All Students Take Calculus!
  - - - Ex. Tanθ = Tan(θ+2pi)
  - - - Ex. Trigonometric functions can also be determined algebraically to be Even or Odd using the same method as highlighted in the turquoise section (1.2) Pic 11
  - - - Transformations
        
        Inverse
        
        the inverse of a base function for sin, cos, tan can be used to solve equations where the theta value is unknown. Essentially it is "arc sin" or on your calculator "-sin" . We can "take the arc sin" of both sides of an equation, which will cancel out the "sin" and leave x to be solve for.
        
        If a function is not one to one it will not have an inverse. However, we can restrict the domain and range to create a similar function which will act as the inverse. Ex. Pic 15 (I have highlighted the change in range)
        
        Controlling Parameters: y=af(b(x+c)+d
        af = vertical stretch or shrink and reflection in x axis (- = reflection)
        b = vertical stretch or shrink and reflection in x axis (- = reflection)
        c = horizontal shift
        d= vertical shift
        
        Sinusoidal: (general Sine function)
        f(x) = A Sin[b(c-x)]+d
        A = amplitude
        B = 2pi/B = Period
        C = Horizontal Shift
        D = Vertical Shift
        (range is D + A)
        
        Reminder: Phase shift, C, in sinusoidal graphs means that the place which the function's wave starts moves over accordingly.
        
        Ex. Pic 10
        
        Ex. Pic 12
- - - - This example asks you to find the cartesian equation for a curve that contains the parametric curve. You would first graph by choosing appropriate t values which are within the limits of the given domain, then you would plot these points. To find the cartesian equation, use the elipses equation. Use your given Y function, and your given X function to fill in the x,y,a, and b slots in the elipses equation. Using this equation eliminates the T value