Summary MHF4U – Power Functions & Polynomials Notes (Advanced Functions)

MHF4U – Power Functions & Polynomials Notes (Advanced Functions)

Review
-​ Function - every x-value produces exactly one y-value
-​ x-value gives two different y values ≠ function
-​ Use vertical line test, a vertical line cannot touch
more than one point on the graph
-​ Some equations may look algebraic but
still fail the vertical line test when graphed
-​ All polynomials are functions, but not all
functions are polynomials
Example
​ red x = (y - 3)2 + 3 [not a function]
blue x = 15

-​ Domain - all possible x-values (independent variable) gives output of real y-values
-​ Range - all possible y-values (dependent variable) derived from the x-values
Example
a)​ y = 2x + 4 ​ where R is real numbers
Domain {X∊R} or (-∞ to + ∞)
Range {Y∊R} or (-∞ to + ∞)
-​ No restriction for domain/range




b)​ f(x)= (x - 3)2
Domain {X∊R} → (-∞ to + ∞)
Range {Y∊R | y ≥ 0} → [0,∞)
-​ No restriction for domain, restriction for range
​



-​Asymptote - a line that a curve gets closer and closer to but never touches
-​ Know there’s an asymptote if the denominator of the function is a higher
degree than the numerator
Example
1
f(x) = 𝑥−3
-​ Vertical asymptote - vertical line (imagine a wall)
-​ Here, x can never = 3 otherwise be dividing by 0
(undefined)
-​ Horizontal asymptote - the horizontal line (imagine a floor)
-​ For this case it reaches close to y = 0 but it never
truly touches it
-​ The bigger the x-value, the larger the denominator,
the closer to 0, but it will never be 0

, MHF4U – Power Functions & Polynomials Notes (Advanced Functions)

Interval Notation
-​ Sets of real numbers may be described in a variety of ways
1)​ As an inequality -3 < x ≤ 5
2)​ Interval (or bracket) notation (-3, 5]​
3)​ Graphically on a numberline
-​ Infinite = ∞ (infinity) or -∞ (negative infinity) → always use a round bracket (cannot
reach these values)
-​ Square bracket → end value is included; Round bracket → end value not included

Polynomial Functions
-​ Power function - simplest type of polynomial function
-​ f(x) = axn​ a → coefficients (a0, a1… an; real numbers)
n → degree of function, (non-negative integer, whole number)
x → variable
-​ Only one term, no other term added
-​ Example: y = -2x4
-​ General form: f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a2x2 + a1x1 + a0
-​ Example: y = 2x3 + 5x2 + 3
-​ Degree of a polynomial - the highest exponent on a variable in an expression
-​ This determines the order how the function is written
-​ Not determined by the leading coefficient
-​ Written in descending order of power (in terms of degrees)
-​ Example: f(x) = x6 + 2x2 + x + 4 has a degree of 6
-​ Leading coefficient (anxn); constant (a0)
-​ Domain is set of real numbers → {X∊R} or (-∞ - to ∞)
-​ Real numbers - all numbers that can be placed on a number line
-​ Range may be all real numbers, but may have an upper/lower bound (not both)
-​ Can have a restriction
-​ No horizontal or vertical asymptotes
-​ Polynomial functions with degree 0 are just horizontal lines (i.e. x = 5)
-​ Shapes of graphs depends on degree of function (generally 5 shapes)
-​ Even degrees → U-shaped, ends go same direction
-​ Odd degrees → S-shaped, ends go opposite directions




Linear; n = 1 Quadratic; n = 2 Cubic; n = 3 Quartic; n = 4 Quintic; n = 5
- Straight line - U-shaped - S-shaped curve - Wider U-shaped - S-shaped curve,
- Always parabola - Always parabola like cubic but
increasing/ - Vertex increasing/ - ‘Flatter’ near steeper near origin
decreasing, no - Symmetrical decreasing vertex - Can increase/
curves​ about y-axis through turn - Symmetrical decrease through
points about y-axis turn points