Parent Functions

Linear Function

f(x) = x

Domain: {x E R}

Range: {y E R}

Quadratic Function

f(x) = x²

Domain: {x E R}

Range: {y E R/0 ≤ y}

Square Root Function

f(x) = √x

Domain: {x E R/0 ≤ x}

Range: {y E R/0 ≤ y}

Reciprocal Function

f(x) = 1/x

Domain: {x E R/x ≠ 0}

Range: {y E R/y ≠ 0}

Asymptote: x = 0, y = 0

Absolute Value Function

f(x) = |x|

Domain: {x E R}

Range: {y E R/0 ≤ y}

Cubic Function

f(x) = x³

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 ( <-- )

  • If negative in bracket (so positive alone), then it moves right ( --> )

Mapping

1. Draw the parent functions’ table of values

2. Create mapping notation using:

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

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

Combinations of Transformations

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