AP PreCalc Unit 1

Know…

• 1.1: Change in Tandem

  • How to write function intervals

    • Ex. 1<x<5

    • Or (1,5) and [1,5]

  • How to find zeroes

    • When line touches the x axis OR when f(x) = 0

  • Concave up and down

    • Bowl UP = Concave Up

    • Bowl DOWN = Concave Down

    • Function could increase or decrease on concave up or down intervals. 

  • Point of inflection

    • Where the graph changes from concave up to concave down or vice versa.

    • In the middle of increasing/decreasing part of concave  up and increasing/decreasing part of concave down. 

  • Y-Intercept

    • What f(0) is equal to OR when the line touches the y axis.

• 1.2: Rates of Change

  • Rate of Change = Slope = (y2-y1)/(x2-x1) 

  • Positive rate of change: Both quantities are the same

  • Negative rate of change: One quantity goes up, one quantity goes down

  • Per indicates rate of change, include in units

  • Average rate of change over interval 

    • Ex. Average rate of change at x=-2 if g(x) = e^x

      • ((e^-1.9)-(e^-2)) / (-1.9+2)

• 1.3: Rates of Change in Linear and Quadratic Functions

  • Average Rate of Change for a Linear Function is always CONSTANT, not necessarily 0. To find, find the slope. 

  • To find the average rate of change of a quadratic function ON A SPECIFIC INTERVAL, find the slope.

  • Average Rate of Change of the Rate of Change for a linear function is ALWAYS ZERO (0). 

  • Average Rate of Change of the Rate of Change for a quadratic function, find f(x) from (-1 -> 2)

    • Ex. x^2+3x+8

      • f(-1) = 6       > 2       >2 

      • f(0) = 8

      • f (1) = 12      > 4      >2 

      • f(2) = 18        >6      

      • ROC of AROC is 2. 

  • To determine if a function is concave up or concave down from a table, see if the RATE OF CHANGE is decreasing or increasing, NOT the f(x). 

• 1.4: Polynomial Function and Rates of Change

  • Local Minima: Any point where a polynomial switches from decreasing to increasing

  • Local Maxima: Any point where a polynomial switches from increasing to decreasing.

  • Global Maximum: Greatest local maxima

  • Global Minimum: Lowest local minima

  • If there are two zeros on a polynomial there is at least ONE local extrema in between the two zeroes. 

    • Ex. if two zeroes are x=-2 and x=4, there is an extrema between -2<x<4/ (-2,4). 

  • If there is an even degree, there is either an absolute minimum and maximum, if it is odd there is none. 

    • If the leading coefficient is negative, there is an absolute maximum. 

    • If the leading coefficient is positive, there is an absolute minimum. 

• 1.5A: Polynomial Functions and Complex Zeros

  • If P(a) = 0 

    • A is a root or zero of P

    • There is an x-intercept at (a,0).

    • If a is a real number, (x-a) is a linear factor of P

  • Multiplicity

    • If n in (x-a)^n is greater than or equal to 2, the zero has a multiplicity of n. 

    • If there is an even multiplicity, the graph “bounces” off the x axis at x=a. 

  • A degree of a polynomial tells you how many zeroes there are, real or nonreal.

  • Non-real zeros: If a + bi is a zero of a polynomial, then its conjugate a-bi is also a zero. 

  • To find where x<0 or where x>0 of a polynomial on a certain interval…

    • Find zeroes

    • Make a table with intervals as x value

    • Sketch (not draw) out a graph of the polynomial

    • Where the function is positive, put a + on the table for that interval and where it is negative or zero, put a - or 0 on the table. 

    • Ex.

  • To find degrees from a table, calculate the differences until they are constant. The difference is the degree. 

• 1.5B: Even and Odd Polynomials

  • An even function is symmetric over x=0

    • To determine analytically if it is even, f(-x) = f(x)

  • An odd function is symmetrical over the origin (0,0)

    • To determine analytically if it is odd, f(-x) = -f(x)

  • If none work, it is neither

• 1.6: End Behavior

  • Limit Notation

    • Lim f(x) = ___

      • x-> - ∞

    • Lim f(x) = ___

      • x->  ∞

• Rational Functions (and Pascal's Triangle) : 1.7-1.11

  • Remember to simplify and find any holes before doing anything. 

  • Domain: Every value x can be set to.

    • To find the domain of a rational function, find the values of x on the denominator and exclude those points from the domain.  

    • Domain Set Notation: {x=∉R | x ≠ 0, x ≠5}

  • Horizontal Asymptotes

    • If the degree on the bottom is greater, the horizontal asymptote is y=0. 

    • If the degree on the top is greater, there is no horizontal asymptote. 

    • If the degrees are the same, the horizontal asymptote is 

y= (Top Leading Coefficient) / (Bottom Leading Coefficient). 

• If there is a horizontal asymptote b, the limit/end behavior for x->  ∞ and x->  ∞ will both go towards the line y=b. 

• To evaluate limits when there is no horizontal asymptote for rational functions… 

• Zeros and Y-Intercept

  • Zeros are found by finding the values of x on the numerator. 

  • Y-Intercept is found by finding f(0). 

• Vertical Asymptotes

  • Vertical asymptotes are found by finding the values of x on the denominator. 

  • Vertical asymptotes are undefined

  • If it cancels out, then it is not a vertical asymptote, it is a hole.

• Holes

  • If a factor (x-a) cancels out fully from the denominator, there is a hole at x=a. 

  • If (c,4) is a hole…

    • = 4

    • =4

  • Holes are undefined

• Pascals Triangle

  • • Used as a shortcut to expand large binomials (ex. (x-7)^5) 

    • Sample Problem: 

• Slant Asymptotes

  • Only happens if there is no horizontal asymptote and the degree of the top is one (1) degree higher than the degree of the denominator. 

  • Found by dividing the rational function and LEAVING OUT THE REMAINDER. 

  • Could either use synthetic or long division. 

• 1.12A: Translations of Functions

  • f(x) + a = function goes up y axis

  • f(x) - a = function goes down y axis

  • f(x-a) = function goes right on x axis

  • f(x+a) = function goes left on x axis

• 1.12B: Dilations of Functions

  • 3f(x) = y value multiplied by 3

  • ⅓ f(x) = y value multiplied by ⅓ 

  • f(3x) = x value multiplied by ⅓

  • f(1/3x) = x value multiplied by 3

• For transformations with both dilations and translations, dilations come first. 

• 1.13: Function Model Selection

  • Linear Function: 1st differences are constant

  • Quadratic Function: 2nd differences are constant

  • Cubic Function: 3rd differences are constant

• 1.14: Function Model Construction

  • Calculator Stat Button

  • Inversely Proportional: y= (k/x^2) 

  • Piecewise Functions