Unit 1 - Precalc

Unit 1

• 1.1

  • VLT (Vertical Line Test) will not always work to determine if a graph is a function or not! (also in AP Testing tips toggle)

  • When graphing, don't connect plotted points since you don’t know if it’s straight/curved/else. (also in AP Testing tips toggle)

  • rate of change is a graph of a function’s slope, but rate of change and slope are NOT interchangeable words

  • rate of change increasing means the function is concave up, since the slope is constantly increasing

  • DON’T abbreviate “rate of change”

  • function → positive , means function is above the x-axis

  • function → increasing , means function has a positive slope/going up

  • function → positive ≠ function → increasing

  • characteristics of rate of change

    roc positive = f(x) increasing

    roc negative = f(x) decreasing

    roc increasing = f(x) concave up

    roc decreasing = f(x) concave down

    • roc positive and increasing: f(x) increasing and concave up ╯

    • roc negative and increasing: f(x) decreasing and concave up ╰

    • roc positive and decreasing: f(x) increasing and concave down ╭

    • roc negative and decreasing: f(x) decreasing and concave down ╮

  • justifying behaviors of graphs (roc rules)

    always use rate of change!

    • increasing f(x) ↗

      rate of change is positive

    • decreasing f(x) ↘

      rate of change is negative

    • concave up (positive parabola) ∪

      The rate of change is increasing

      the function’s rate of change is increasing over consecutive equal length intervals so the function is concave up

    • concave down (negative parabola) ∩

      The rate of change is decreasing

      the function’s rate of change is decreasing over consecutive equal length intervals so the function is concave down

    • quadratic

      over consecutive equal-length input-value intervals of size , there is a constant second difference in output values of

    • linear

      rate of change is constant

      over consecutive equal-length input-value intervals of size , there is a constant difference in output values of

• 1.2

  • average rate of change

    • solve by

      1. find the given interval

      2. find the point associated with the endpoints of that interval

      3. use slope formula to find the average rate of change between those two points

         $\frac {y_2 - y_1} {x_2 - x_1 }$

  • instantaneous rate of change

    • same thing as finding the rate of change of a function at a given point

    • we only approximate the rate of change at a point because we are not at calculus level yet

    • find by using the average rate of change over small intervals (eg. [1,2] or [1,1.001] or [0.999,1] )

• 1.3

  • tangent = intersecting in only one spot (touches)

  • secant = intersecting in 2 or more distinct spots

  • a tangent line can represent the rate of change at one point

  • a secant line can represent the average rate of change of a larger interval, where the line intersects the function

  • roc for linear functions

    • if f(x) is linear, the rate of change is a constant, horizontal line

      f(x) =↗

      roc = →

    • justify by…

      the function is linear because the rate of change is constant

      over consecutive equal-length input-value intervals of size , there is a constant difference in output values of _.

  • roc for quadratic functions

    • if f(x) is quadratic, the rate of change is an increasing/decreasing line

      f(x) = ∪

      roc = ↗

      or

      f(x) = ∩

      roc = ↘

    • justify quadratics by…

      over consecutive equal-length input-value intervals of size , there is a constant second difference in output values of

  • when the roc graph looks like a parabola, f(x) is cubic

  • find the roc / slope / degree / concavity with a chart of points

    1. with the x and y values, subtracting each number by the one above it

    2. divide the y/x. this is the slope / roc

       use the roc rules to find what type of function it is

    3. to find the degree, keep subtracting the y-values until all the results are equal (could work the first time)

    4. the number of rounds it takes to get all the subtracted y-values equal, is the degree.

• 1.4

  • polynomial functions

    • degree is a positive integer

    • leading coefficient is a real number

  • types of extrema

    • relative/local extrema

      • where the polynomial switches between increasing and decreasing or has an endpoint

      • local max/min

    • absolute/global extrema

      • the local extrema with the greatest/least y values

      • absolute max/min

    • notated with:

      local max/min at x=___(x-values)

      absolute max/min = _(y-value) at x=(x-value)

  • extrema/end behavior on polynomials

    • even degree→same end behaviors on both sides

    • odd degree→opposite end behaviors

    • 2 zeroes→must be at least 1 local max/min between them

    • even degree→must have a global max/min

  • points of inflection

    • is when a function changes between concave up to concave down

    • at this point, the rate of change of changes between increasing to decreasing

    • is an extrema in the roc graph

    • located in the middle of two neighboring extrema

  • in interval notation, when asking for increasing/decreasing part of a function, it is [] unless otherwise noted

  • in interval notation, when asking for when it is concave up/down, it is () unless otherwise noted

• 1.5

  • if a polynomial has a degree n > 0, it will have a total of n zeroes or roots (may be nonreal)

  • x-intercepts are only real zeroes! zeroes and roots can be nonreal!

  • if (r,0) is an x-intercept, then r is a root of the polynomial f(x) and r is a solution of the equation f(x)=0

    these are different!

  • remainder theorem

    • when we divide one polynomial (the dividend) by another (the divisor), we obtain a quotient polynomial and a remainder

      (quotient)(divisor)+remainder=dividend

    • when f(x) is divided by (x-c), then the remainder is f(c)

  • factor theorem

    • if the divisor is (x-c) and the remainder is f(c),

      then if f(c)=0, x-c is a factor

      if x-c is a factor, then f(c)=0

  • multiplicity

    • if (x-a) is repeated n times, the corresponding zero has a multiplicity of n

    • odd multiplicity→polynomial passes through the axis at the point of the zero

      multiplicity of 1→goes through like a line

      multiplicity > 1→goes through changing concavity

    • even multiplicity→polynomial touches the axis at the point of the zero

  • nonreal roots come in pairs!

    if a-bi is a root, a+bi is also a root!

  • f(x)=p(x+q)(x+r)(x+s), $ax^2+bx+c$

    pqr*s = c (y-intercept)

  • f(x)>0 asks when f(x) is above the x-axis

    f(x)<0 asks when f(x) is below the x-axis

    f(x)=0 asks when f(x) is on the x-axis

  • to solve nonlinear inequalities, you can use a sign chart or find the zeroes and visualize the graph

  • symmetry

    • even functions are symmetric over the y-axis (when they aren’t shifted)

      f(-x)=f(x)

      (x,y)=(-x,y)

      eg. $x^2,3x^4,-5$

    • odd functions are symmetric about the origin (when they aren’t shifted)

      f(-x)=-f(x)

      (x,y)=(-x,-y)

      eg. $x^3,3x,-5x^9$

    • you can find if a function has an odd or even degree by finding f(-x) and seeing if they are equal to f(x) (even) or -f(x) (odd). if they don’t match, they are probably shifted

    • y=0 is the only function that is both even and odd

• 1.6

  • end behavior

    • describes how a function behaves as it moves infinitely to the right and left

    • what happens to the y-values as x increases or decreases without bound

    • justified by writing:

      As the x-values increases/decreases without bound, the y-values of f(x) increase/decrease without bound.

    • notated in limit notation:

      
      

      

    • right side:

      leading coefficient is positive (+) →end behavior goes up/increases

      leading coefficient is negative (-) → end behavior goes down/decreases

    • left side:

      degree is even → same as the right side

      degree is odd → opposite of the right side