11-03: Various Types of Functions

Parent Functions

Linear Function

f(x) = x

xy
-2-2
-1-1
00
11
22

Domain: {x E R}

Range: {y E R}

Quadratic Function

f(x) = x²

xy
-24
-11
00
11
24

Domain: {x E R}

Range: {y E R/0 ≤ y}

Square Root Function

f(x) = √x

xy
00
11
42

Domain: {x E R/0 ≤ x}

Range: {y E R/0 ≤ y}

Reciprocal Function

f(x) = 1/x

xy
-2-1/2
-1-1
-0.5-2
0.52
11
21/2

Domain: {x E R/x ≠ 0}

Range: {y E R/y ≠ 0}

Asymptote: x = 0, y = 0

Absolute Value Function

f(x) = |x|

xy
-22
-11
00
11
22

Domain: {x E R}

Range: {y E R/0 ≤ y}

Cubic Function

f(x) = x³

xy
-2-8
-1-1
00
11
28

Domain: {x E R}

Range: {y E R}

Transformations of Parent Functions

Transformed functions: @@f(x) = a(k(x-d)) + c@@

  • Vertical Stretch: ==a==
      * By a factor of….
      * If negative, reflection in the x axis
  • Horizontal Stretch: ^^k^^
      * Always 1/k (flipped)
      * By a factor of….
      * If negative, reflection in the y axis
  • Vertical Translation: ==c==
      * if positive, moves up
      * If negative, moves down
  • Horizontal Translation: ^^d^^
      * Always the opposite sign of what it is in the brackets (sign is flipped)
      * If positive in bracket (so negative alone), then it moves left (

Mapping

  1. Draw the parent functions’ table of values
  2. Create mapping notation using:

Mapping Notation: @@((1/k)x + d, ay + c)@@

  1. Apply mapping notation to the parent function and graph (following BEDMAS, order of operations)

Combinations of Transformations

Quadraticg(x) = a(k(x-d))² + c
Reciprocalg(x) = a(1/(k(x-d)) + c
Cubicg(x) = a(k(x-d))³ + c
Square Rootg(x) = a(√k(x-d) ) + c
Absolute Valueg(x) = a |k(x-d)| + c

Domain and Range of Functions from Equations

  • Linear:
      * D: {x E R}
      * R: {y ER}
  • Cubic:
      * D: {x E R}
      * R: {y ER}
  • Quadratic:
      * D: {x E R}
      * R: {y E R/0 ≤ y}
        * c is the restriction (replacing zero)
  • Absolute Value:
      * D: {x E R}
      * R: {y E R/0 ≤ y}
        * c is the restriction (replacing zero)
  • Reciprocal:
      * D: {x E R/ x ≠ 0}
        * c replaces the restriction
      * R: {y E R/ y ≠ 0}
        * d replaces the restriction
          * The function cannot touch the asymptote thus the asymptote is our restriction
  • Square Root:
      * D: {x E R / 0 ≤ x}
        * Make the number under the square root sign as small as it can be, so zero (because it cannot be negative since you can’t have a negative radicand)
      * R: {y E R / 0 ≤ y}
        * Look at what your lowest y could be as a result of the reduction of x for domain

The Inverse Function

  1. Write the function in x-y notation
  2. Swap x and y
  3. Solve for y