Advanced functions CPT
Prince Dhillon
Advanced functions
MHF4U
Submission date: January 16, 2022
Table of contents page
Polynomial…………………………………………………………………………………………………………………………3
Rational……………………………………………………………………………………………………………………………..5
Logarithmic………………………….…………………………………………………………………………………..………7
Exponential………………………………………………………………………….……………………………..…….……..9
Trigonometric………………………………………………………………………………………………………………….11
Combinations…………………………………………………………………………………………………………….……14
References……………………………………………………………………………………………………………….………16
Polynomial
Transformations:
Vertical stretch by a factor of 3
Horizontal translations by 2 units left
Vertical translation 1 unit down
Domain: {x ∈ R}
Range: {y ∈ R | y ≥ -1}
Continuity: {x ∈ R}
End behaviour: x → ∞, y → ∞
x → -∞, y → ∞
Increasing: [-1, ∞)
Decreasing: (-∞, -1]
Maximum: None
Minimum: (-2, -1)
Horizontal Asymptote: None
Vertical Asymptote: None
Red: f(x) = 3(x + 2)^2 - 1
Blue: Parent = x^2
Symmetry: None
Main points: (-3, 2), (-2, -1), (-1, 2)
Polynomial example:
Calculating a firm's revenue, where the y
value represents profit and the x value
is the number of days since the
business began. It informs the business
owner of his potential earnings in x
days.
Rational
Transformations:
Horizontal translations by 2 units right
Vertical translation 2 units up
Domain: {x ∈ R | x ≠ -2, 2}
Range: {y ∈ R | y ≠ 1.75, 2}
Continuity: Hole discontinuity at x = -2
Infinity discontinuity at x = 2
End behaviour: x → ∞, y → 2
x → -∞, y → 2
Increasing: None
Decreasing: (-∞, 2), (2, ∞)
Maximum: None
Minimum: None
Horizontal Asymptote: y = 2
Vertical Asymptote: y = 2
Red: g(x) = ((x + 2) / (x^2 - 4)) + 2
Blue: Parent = 1/x
Symmetry: None
Main points: (0, 1.5), (1.5, 0), (1, 1), (2.5, 4), (3, 3), (4, 2.5)
Rational example:
Calculating physics equations, such
as v = d/t for velocity. D and t are
sometimes in relation to another
value, or they can both be polynomial
functions.
Logarithmic
Transformations:
Horizontal compression by a factor of 2
Horizontal translations by 1 units right
Domain: {x ∈ R | x > 1}
Range: {y ∈ R}
Continuity: {x ∈ R | x > 1}
End behaviour: x → ∞, y → ∞
x → 1-, y → -∞
Increasing: (1, ∞)
Decreasing: (∞, 1)
Maximum: None
Minimum: None
Horizontal Asymptote: None
Vertical Asymptote: x = 1
Red: h(x) = log(2x - 2)
Blue:Parent = log(x)
Symmetry: None
Main points: (1.5, 0), (2, 0.3), (3, 0.6)
Logarithmic example:
In chemistry, determining the basicity
or acidity of a solution. Because the
pH scale is built on a logarithmic
scale with a base of 10, each pH value
represents a full level of Hydrogen
that may be described using
logarithmic functions.
Exponential
Transformations:
Horizontal translations by 2 units left
Vertical translation 1 unit up
Domain: {x ∈ R}
Range: {y ∈ R | y > 1}
Continuity: {x ∈ R}
End behaviour: x → ∞, y → ∞
x → -∞, y → 1
Increasing: (-∞, ∞)
Decreasing: (∞, -∞)
Maximum: None
Minimum: None
Horizontal Asymptote: y = 1
Vertical Asymptote: None
Blue: j(x) = 2^x+2 + 1
Green: Parent = 2^x
Symmetry: None
Main points: (-2, 2), (0, 5), (1, 9)
Exponential example:
In biology, calculate the population of
bacteria. After knowing a bacteria's
half life or double life, an exponential
function may be used to compute the
population after a particular length of
time x.
Trigonometric
Transformations:
Vertical stretch by a factor of 2
Horizontal translations by 3 units left
Vertical translation 1 unit down
Domain: {x ∈ R}
Range: {y ∈ R | -3 ≤ y ≤ 1}
Continuity: {x ∈ R}
End behaviour: None
Increasing: None
Decreasing: None
Maximum: (360° + 360°k, 1) where k ∈ Z
Minimum: (180° + 360°k, -3) where k ∈ Z
Horizontal Asymptote: y = -3, 1
Purple: k(x) = 2sin(x + 90°) - 1
Black: parent= sin(x)
Vertical Asymptote: None
Symmetry: Even
Main points: (0°, 1), (180°, -3), (360°, 1)
Trigonometric example:
Based on time x, calculate the height
of a human on a ferris wheel. The
cycles reflect the ferris wheel's round
movement, which may be transferred
to a trigonometric function.
Composite function
In most cases, a composite function is a
function that is written inside of another
function. The process of composing a
function entails replacing one function for
another.
Combinations
Function:
3(x + 2)^2 - 1 + ((x + 2) / (x^2 - 4)) + 2
Domain: {x ∈ R | x ≠ 2}
Range: {y ∈ R | y ≠ 2}
Continuity: Infinite discontinuity at x = 2
End behaviour: x → ∞, y → ∞
x → -∞, y → ∞
Increasing: [2.2, ∞), (-2, 1.8]
Decreasing: (-∞, -2), (1.8, 2), (2, 2.2)
Maximum: None
Minimum: None
Horizontal Asymptote: None
Vertical Asymptote: x=2
Symmetry: None
Main points: (-2, undefined), (2, 59), (1.8, 40)
Blue: 3(x + 2)^2 - 1 + ((x + 2) / (x^2 - 4)) +
Blue: m(x) = f(x) + g(x)
Combinations example:
When you need to add, subtract,
multiply, divide, or sub in two related
functions to examine how one affects
the other, combination functions come
in handy.
References:
Graphing calculator. Desmos. (n.d.). Retrieved December 16, 2021, from
https://www.desmos.com/calculator